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Integration by parts of (e^x)(cos x)

Video vocabulary

situation

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US / ˌsɪtʃuˈeɪʃn /

UK / ˌsɪtʃuˈeʃən /

A1
  • noun
  • Place, position or area that something is in
  • Circumstance or condition surrounding an event
  • An unexpected problem or difficulty

well

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US / wel /

UK / wɛl /

A1
  • adjecitve
  • Being in a fortunate, good condition or state
  • Healthy and in good condition
  • adverb
  • In an acceptable, proper, or successful manner
  • In a complete, total, or full manner
  • A great and considerable degree
  • In a successful way, in a good or satisfactory way
  • exclamation
  • Used to begin or continue a particular statement
  • Used to show you want to start saying something
  • Used to express that you are waiting for a reply
  • You say this to change the topic of a conversation
  • Used to indicate you are making a final remark
  • noun
  • Long deep hole in the ground to get water
  • verb
  • To cause your eyes to fill with tears

just

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US / dʒʌst /

UK / dʒʌst /

A1
  • adjecitve
  • Proper or appropriate; as is deserved
  • Doing or being what is right or fair
  • Legally correct; having enough evidence
  • adverb
  • Almost; nearly
  • Exactly at the time of
  • Exactly (the same as, like)

equation

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US / ɪˈkweɪʒn /

UK / ɪˈkweʒən, -ʃən /

B1
  • noun
  • Mathematical statement showing things to be equal
  • Complex set of different facts, ideas or issues

define

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US / dɪ'faɪn /

UK / dɪˈfaɪn /

A2
  • verb
  • To explain the meaning of words
  • To set or mark the limits of something

definitely

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US / ˈdefɪnətli /

UK / ˈdefɪnətli /

A2
  • adverb
  • Without question; beyond doubt

arbitrarily

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US / 'ɑ:bɪtrəlɪ /

UK / ˌɑrbəˈtrɛrəlɪ /

B1
  • adverb
  • Without using reason, often unfairly; on a whim

time

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US / taɪm /

UK / taɪm /

A1
  • noun
  • Speed at which music is played; tempo
  • Point as shown on a clock, e.g. 3 p.m
  • Number of hours, minutes needed to do something
  • Occasion when something happens
  • Period or occasion that something occurred
  • Period in history or the past
  • Something measured in minutes, hours, days, etc.
  • How long an event takes; duration
  • verb
  • To check speed at which music is performed
  • To choose a specific moment to do something
  • To measure how long an event takes, e.g. a race
  • To schedule something to occur at a specific moment

set

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US / set /

UK / sɛt /

A1
  • adjecitve
  • Prepared for something; ready
  • noun
  • Complete group of something
  • Place where a television show or movie is filmed
  • Group of games in a tennis match
  • Device that receives radio or television signals
  • verb
  • To make a clock state or ring at a particular time
  • To decide upon or choose something
  • To make something ready for use (e.g. table)
  • To decide on a price for something
  • To become hard (of glue, concrete etc.)
  • To be located in a specific place or time
  • To put (e.g. a vase) down carefully in a place
  • To put someone in a certain condition or state
  • (Of the sun) to go lower than the horizon

positive

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US / ˈpɒzətɪv /

UK / ˈpɑzɪtɪv /

A2
  • adjecitve
  • Showing agreement or support for something
  • Being sure about something; knowing the truth
  • Having the charge produced by electrons
  • Being good or useful
  • Showing that a chemical is present in a test

why

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US / waɪ /

UK / hwaɪ, waɪ /

A1
  • adverb
  • Question word to ask for a reason
  • exclamation
  • (Used to express surprise or approval)

begin

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US / bɪ'ɡɪn /

UK / ˈbeɡɪn /

A1
  • verb
  • To do the first part of an action; to start

few

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US / fju: /

UK / fju /

A1
  • pronoun
  • Not many, a small number

this

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US / ðɪs /

UK / ðɪs /

A1
  • adjecitve
  • Used to indicate something already been discussed
  • Used to indicate something happening around now
  • adverb
  • An indication by a physical motion or gesture
  • A specific amount of
  • determiner
  • Person, thing, or idea near you
  • pronoun
  • Person, thing that has already been discussed

term

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US / tɜ:m /

UK / tɚm /

A1
  • noun
  • Conditions applying to an agreement, contract
  • Length of time something is expected to happen
  • Fixed period of weeks for learning at school
  • The (precise) name given to something
  • verb
  • To call; give a name to

less

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US / les /

UK / lɛs /

A1
  • adjecitve
  • Smaller in amount or degree
  • adverb
  • To a smaller degree
  • noun
  • A smaller amount
  • preposition
  • With the subtraction of

comparable

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US / ˈkɒmpərəbl /

UK / ˈkɑmpərəbəl /

A2
  • adjecitve
  • Being similar in some way

appreciably

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US / əˈpriːʃəbli /

UK / əˈpriːʃəbli /

C1
  • adverb
  • Considerably; noticeably

you

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US / jʊ /

UK / ju /

A1
  • pronoun
  • Person someone is speaking or writing to
  • Person or people in general

little

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US / 'lɪtl /

UK / ˈlɪtl /

A1
  • adjecitve
  • Being short in length of time
  • Not great in size, amount, or degree; small
  • Being very young or younger than others
  • adverb
  • By an amount or degree that is not very much
  • Not often; rarely
  • noun
  • A person's name

one

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US / wʌn /

UK / wʌn /

A1
  • pronoun
  • You; we; people in general; someone
  • The thing or person previously referred to
  • adjecitve
  • Used to refer to people generally
  • Being a strong example of (something mentioned)
  • number
  • Number 1

other

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US / 'ʌðə(r) /

UK / ˈʌðər /

A1
  • adjecitve
  • A thing different from that mentioned
  • determiner
  • (Something) else; not the first (one)
  • pronoun
  • Being the one and only thing of that category

sal

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US / sæl /

UK / sæl /

A1
  • noun
  • A North Indian tree that yields hard, durable timber and dammar resin.

average

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US / 'ævərɪdʒ /

UK / ˈævərɪdʒ, ˈævrɪdʒ /

A2
  • adjecitve
  • Typical or normal; usual; ordinary
  • noun
  • Total of numbers divided by the number of items
  • verb
  • To add numbers then divide by the number of items

able

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US / 'eɪbl /

UK / ˈebəl /

A2
  • adjecitve
  • Skilled; good at something
  • Having the ability to do something

might

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US / maɪt /

UK / maɪt /

A1
  • other
  • Used to politely request an action
  • Used to express that something could happen
  • Power, strength, force or influence

than

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US / ðən /

UK / ðən /

A1
  • conjunction
  • Used to compare two things that are not equal

how

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US / haʊ /

UK / haʊ /

A1
  • adjecitve
  • To what degree? in what amount?
  • adverb
  • (Used to explain the way to do something)
  • exclamation
  • For what purpose or reason?
  • (Used to ask someone or something's condition)

kind

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US / kaɪnd /

UK / kaɪnd /

A1
  • adjecitve
  • In a caring and helpful manner
  • friendly and considerate
  • noun
  • One type of thing

the

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US / ðə /

UK / ðə /

A1
  • article
  • Used to refer to something already mentioned
  • Used to show there is only one of something
  • determiner
  • Used to describe all of a family

let

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US / let /

UK / lɛt /

A1
  • verb
  • To allow someone to do something
  • Introducing a suggestion to do something together
  • To rent a house, etc. to others

try

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US / traɪ /

UK / traɪ /

A1
  • noun
  • Act of putting the ball over the rugby goal line
  • verb
  • To make an effort, to attempt to do something
  • To examine someone or something in a court of law
  • To test to see if it is good or you like it
  • To make a person go through hardship; test
  • To examine something to see if it works

guess

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US / ɡes /

UK / ɡɛs /

A1
  • noun
  • Answer given when you don't know if it is correct
  • verb
  • To give an answer without knowing if it is correct

too

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US / tu: /

UK / tu /

A1
  • adverb
  • In addition, also; with emphasis
  • Say that something is more than you want

somehow

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US / ˈsʌmhaʊ /

UK / ˈsʌmˌhaʊ /

A1
  • adverb
  • In a way that is not known or certain

divide

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US / dɪ'vaɪd /

UK / dɪˈvaɪd /

A2
  • verb
  • To split numbers by another number, e.g. 6 / 2 = 3
  • To separate something into equal pieces

complicated

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US / 'kɒmplɪkeɪtɪd /

UK / ˈkɑmplɪˌketɪd /

B1
  • adjecitve
  • Difficult to do, understand or use
  • verb
  • To make more difficult to do, understand or use

here

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US / hɪə(r) /

UK / hɪr /

A1
  • adverb
  • In this place; at this location, position or time

either

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US / ˈaɪðə(r) /

UK / ˈiðɚ, ˈaɪðɚ /

A1
  • adjecitve
  • One of two (used when there is a choice of two)
  • adverb
  • (After negative statements) likewise, in addition
  • conjunction
  • Used as the first of two connected things
  • determiner
  • Used as the first of two connected things
  • pronoun
  • The one or the other

side

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US / saɪd /

UK / saɪd /

A1
  • adjecitve
  • In addition to the central or most important thing
  • Next to, or very close to, the right or left of
  • noun
  • Position or opinion that is opposite to another
  • Right or left part of a person's body
  • Feature of someone's personality
  • Straight edge of an object
  • One of two or more groups fighting each other
  • verb
  • To decide to agree with one point, not the other

integration

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US / ˌɪntɪ'ɡreɪʃn /

UK / ˌɪntɪˈɡreʃən /

B1
  • noun
  • Act of bringing things together into a whole

integral

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US / ˈɪntɪgrəl /

UK / ˈɪntɪɡrəl, ɪnˈtɛɡrəl /

B1
  • adjecitve
  • Being an essential part of something; necessary

dramatically

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US / drə'mætɪklɪ /

UK / drəˈmætɪkl:ɪ /

B1
  • adverb
  • In a sudden or extreme manner

essentially

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US / ɪˈsenʃəli /

UK / ɪˈsenʃəli /

A2
  • adverb
  • Basically; (said when stating the basic facts)

happen

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US / 'hæpən /

UK / ˈhæpən /

A1
  • verb
  • To take place or occur

get

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US / ɡet /

UK / ɡɛt /

A1
  • verb
  • To become affected by illness or disease
  • To begin to feel or understand an emotion or idea
  • To send or bring someone or something home, etc.
  • To go somewhere to obtain something
  • To (cause to) do a particular thing
  • To obtain, receive or be given something
  • To currently have
  • To prepare for use
  • To understand something being said or read

what

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US / wɒt /

UK / hwɑt, hwʌt, wɔt, wʌt,hwət, wət /

A1
  • pronoun
  • Thing or things that; the same kind as
  • Used to emphasize something you are about to say
  • adjecitve
  • Used to refer to a particular amount, or number
  • Used to place emphasis on a particular quality
  • adverb
  • To a certain degree
  • determiner
  • Question words used for asking for information
  • exclamation
  • Used to express excitement, shock, or surprise
  • Used to ask people to repeat something they said

notation

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US / nəʊ'teɪʃn /

UK / noˈteʃən /

C2
  • noun
  • Set of written signs are used in music or math

again

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US / ə'ɡen /

UK / əˈɡɛn /

A1
  • adverb
  • One more time, once more

solve

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US / sɒlv /

UK / sɑ:lv /

A2
  • verb
  • To figure out a way to handle a difficult problem

interesting

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US / ˈɪntrəstɪŋ /

UK / ˈɪntrɪstɪŋ, -tərɪstɪŋ, -təˌrɛstɪŋ /

A1
  • adjecitve
  • Taking your attention; making you want to know
  • verb
  • To make someone want to know about something
  • To persuade to do, become involved with something

hard

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US / hɑ:d /

UK / hɑː(r)d /

A1
  • adjecitve
  • Difficult to do; difficult to understand
  • (Of facts) not able to be misunderstood; clear
  • (Of alcohol) strong
  • Involving or requiring lots of work, effort, care
  • Being full of pain, trouble, and problems
  • Strong
  • (Of edge) clear; sharp
  • Difficult to bend, break or cut; solid
  • adverb
  • With lots of force, power, or impact

that

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US / ðæt /

UK / ðæt, ðət /

A1
  • adjecitve
  • The person or thing being talked about
  • Used to identify something both the speakers know
  • adverb
  • Used to reinforce adjectives and adverbs
  • To such a degree
  • conjunction
  • Used to connect a noun clause
  • Used to introduce an adverbial clause
  • Used to introduce a noun clause
  • determiner
  • Used as a noun to refer to something
  • pronoun
  • Referring an object far away
  • Used to refer to the relative pronoun ‘which’

select

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US / sɪ'lekt /

UK / sɪˈlɛkt /

A2
  • verb
  • To choose a suitable thing from a group

care

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US / keə(r) /

UK / kɛr /

A1
  • noun
  • Treatment you receive for your health or safety
  • Something that a person worries about
  • verb
  • To feel interest, concern, or worry
  • To look after someone or something

second

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US / 'sekənd /

UK / ˈsɛkənd /

A1
  • adjecitve
  • 2nd; the one following first
  • noun
  • Unit of time; one sixtieth of a minute
  • Short period of time that you wait for someone
  • (Do) soon

then

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US / ðen /

UK / ðɛn /

A1
  • adverb
  • A time that is not now
  • Following another thing in time or arrangement
  • Following on reasonably from what was just stated
  • At that time, not now

twice

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US / twaɪs /

UK / twaɪs /

A1
  • adverb
  • Two times

right

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US / raɪt /

UK / raɪt /

A1
  • adjecitve
  • Correct or true
  • The south side of you looking at the rising sun
  • adverb
  • Exactly in a place or time
  • Being appropriate for a particular event
  • In a straight or direct manner
  • noun
  • Something you are legally or morally allowed to do
  • Ideal of what is just and good
  • Political view based on conservatism
  • verb
  • To fix something previously wrong or unjust
  • To make upright, as a boat that has turned over

like

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US / laɪk /

UK / laɪk /

A1
  • adjecitve
  • Being equal in status or kind to something else
  • Similar; the same
  • adverb
  • For example
  • preposition
  • Used as a filler; used to ad emphasis
  • In a way similar to the way something is done
  • Such as; for example
  • verb
  • To want to
  • To find something pleasing; to prefer something

similarly

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US / ˈsɪmələli /

UK / ˈsɪmələrli /

A2
  • adverb
  • In a way almost the same as something else

about

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US / ə'baʊt /

UK / əˈbaʊt /

A1
  • adverb
  • To be ready to; going to
  • Nearly; approximately; roughly
  • Near; next to; close to
  • In every direction; in the area of
  • preposition
  • Concerning or of (a topic)

both

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US / bəuθ /

UK / boθ /

A1
  • determiner
  • Used to refer to two things at the same time
  • pronoun
  • Used to refer to two things at the same time

expression

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US / ɪk'spreʃn /

UK / ɪkˈsprɛʃən /

A2
  • noun
  • Act of making your thoughts and feelings known
  • Group of words that have a specific meaning

all

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US / ɔ:l /

UK / ɔl /

A1
  • adverb
  • Completely; totally
  • determiner
  • Being every one of something
  • pronoun
  • 100% of something; the complete amount

original

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US / ə'rɪdʒənl /

UK / əˈrɪdʒənəl /

A2
  • adjecitve
  • Being first made, thought or performed; fresh
  • Being unique or thinking differently from others

neither

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US / ˈnaɪðə(r) /

UK / ˈniðɚ, ˈnaɪ- /

A2
  • adjecitve
  • Not one or the other not either none of two things
  • adverb
  • Similarly not, also not
  • conjunction
  • Not in either case
  • determiner
  • Not one or the other
  • pronoun
  • Not either one

derivative

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US / dɪˈrɪvətɪv /

UK / dɪˈrɪvətɪv /

B1
  • adjecitve
  • Not from original ideas; from somewhere else

bit

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US / bɪt /

UK / bɪt /

A1
  • noun
  • Device put in a horse's mouth to control it
  • Small piece of something
  • verb
  • (E.g. of fish) to take bait and be caught

take

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US / teɪk /

UK / tek /

A1
  • noun
  • Filming of a single scene for TV or the movies
  • verb
  • To accept something that is offered
  • To use a method of traveling, e.g. a bus or train
  • To use a camera to make (a photo)
  • To pick up something and go away with it
  • To study a subject in school
  • To attempt to pass (an exam)
  • To require a certain amount of time, money, space

its

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US / ɪts /

UK / ɪts /

A1
  • adjecitve
  • Referring to or belonging to ‘it’
  • determiner
  • Of something; belonging or related to something
  • other
  • Possessive form of “it”

rewrite

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US / ˌri:'raɪt /

UK / riˈraɪt /

A2
  • verb
  • To write again to improve, or correct errors

figure

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US / ˈfiɡə /

UK / ˈfɪɡjɚ /

A1
  • noun
  • Your body shape
  • Numbers in a calculation
  • Doll-like thing meant to represent a person
  • Picture or diagram giving information in a text
  • Person who is very important or famous
  • Shape of a person seen indistinctly or in outline
  • Amount that is expressed in numbers
  • verb
  • To appear in a game, play or event
  • To calculate how much something will cost
  • To understand after thinking; work out

they

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US / ðeɪ /

UK / ðe /

A1
  • pronoun
  • Two or more people, animals, or things

subtract

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US / səbˈtrækt /

UK / səbˈtrækt /

B2
  • other
  • To take away a value or amount from another
  • verb
  • To take away a value or amount from another

look

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US / lʊk /

UK / lʊk /

A1
  • noun
  • Person's beauty or physical attractiveness
  • Particular style of dress, clothes, or appearance
  • The way someone, something appears to be
  • Act of searching for or examining something
  • verb
  • To appear to be; seem

simple

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US / 'sɪmpl /

UK / ˈsɪmpəl /

A1
  • adjecitve
  • Not hard to understand or do; not complex
  • Having only a few parts; not complex
  • Being ordinary or normal
  • Being plain; without a lot added for style

add

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US / æd /

UK / æd /

A1
  • verb
  • To include as well
  • To put two things or numbers together, e.g. 2+2

blue

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US / blu: /

UK / blu /

A1
  • adjecitve
  • Color of the clear sky
  • Feeling sad or down

our

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US / 'aʊə(r) /

UK / aʊr /

A1
  • determiner
  • Belonging to us

copy

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US / ˈkɒpi /

UK / ˈkɑ:pi /

A1
  • noun
  • A counter for items that are the same
  • Fake product pretending to be an original
  • An imitation of an original
  • A duplicate of a paper printed in a photocopier
  • Written text for printing, use in ads, etc.
  • verb
  • To make something look the same as the original
  • To act in the exact same way as someone else
  • To read and then write down what you read

now

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US / naʊ /

UK / naʊ /

A1
  • adverb
  • At the present time or moment
  • Right away or in the next possible moment
  • Used to refer to something that just happened
  • Time between the present and a moment in the past
  • conjunction
  • Resulting from or because of something

whole

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US / həul /

UK / hol /

A1
  • adjecitve
  • Complete or full; all of
  • Being in a good condition
  • noun
  • Complete amount or sum; total

neat

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US / ni:t /

UK / nit /

A2
  • adjecitve
  • That you approve of; cool
  • (Of alcohol) not weakened with water or a mixer
  • Not messy; clean and orderly; simple and effective

can

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US / kæn /

UK / kən,kæn /

A1
  • noun
  • A metal container for storing food
  • other
  • Able to; have the skill to; have the time to
  • verb
  • To have permission to
  • To be possible to do
  • To preserve food by sealing it in a metal container

not

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US / nɒt /

UK / nɑt /

A1
  • adverb
  • Word indicating the negative
  • (Used to form the negative of verbs)

distribute

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US / dɪˈstrɪbju:t /

UK / dɪˈstrɪbjut /

A2
  • verb
  • To hand out or give to people
  • To send products to stores to sell
  • To spread something over an area

business

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US / ˈbɪznəs /

UK / ˈbɪznɪs /

A1
  • noun
  • A company formed for making profit
  • Matter that has to be dealt with; task; situation
  • Activity of working to make a profit

but

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US / bət /

UK / bʌt,bət /

A1
  • conjunction
  • Used before you say something different, opposite

apply

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US / ə'plaɪ /

UK / əˈplaɪ /

A1
  • verb
  • To spread a substance or liquid over a surface
  • To commit your time and effort to doing something
  • To make something useful in a certain situation
  • To be relevant for a situation
  • To ask formally for (job, permission etc.)

become

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US / bɪ'kʌm /

UK / bɪˈkʌm /

A1
  • verb
  • To begin to be; grow to be; develop into
  • To be suitable for a particular job or role

want

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US / wɔnt /

UK / wɑnt, wɔnt /

A1
  • noun
  • Something you desire or wish for
  • State or condition of poverty
  • verb
  • To desire or wish for something; hope for a thing

equal

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US / 'i:kwəl /

UK / ˈikwəl /

A1
  • adjecitve
  • Same in shape, size, or number
  • verb
  • To add up or be the same as

give

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US / ɡɪv /

UK / ɡɪv /

A1
  • noun
  • Degree of flexibility in something, a material
  • verb
  • To hand over or present something to someone
  • To cause someone to have or experience something

whoop

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US / wu:p /

UK / hup, hwup, wup /

B2
  • exclamation
  • Loud cry expressing excitement or happiness
  • verb
  • To shout loudly in a happy or excited manner

color

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US / 'kʌlə(r) /

UK / ˈkʌlɚ /

A1
  • noun
  • Quality of things you can see, e.g. red, blue
  • Pink or red in your face, e.g. after being ill
  • The shade of someone's skin e.g. black, or yellow
  • verb
  • To change or affect someone's opinion
  • To make something colorful using colored pencils

round

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US / raʊnd /

UK / raʊnd /

A1
  • adjecitve
  • Being in the shape of a circle or ball
  • Consisting of curved and not straight lines
  • Somewhat fat or curved in physical appearance
  • Being a whole number, not a fraction, decimal
  • adverb
  • From beginning to end (e.g. of the year)
  • noun
  • One of the three-minute periods in boxing
  • Drinks bought by one person for a group
  • Series regular visits, routes, or stops at places
  • Single shot or bullet fired from a gun
  • An object with a circular shape
  • One stage in a series of games, e.g. football
  • One's turn in a game such as cards
  • preposition
  • Going around
  • verb
  • To cause or make a circular or curved shape
  • To go or pass around the bend or corner
  • To change from a fraction to nearest whole number

same

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US / seɪm /

UK / sem /

A1
  • adjecitve
  • Being equal in quality and amount
  • Identical; not different
  • Not changing; constant

part

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US / pɑ:t /

UK / pɑ:rt /

A1
  • adjecitve
  • To some degree; half; not fully
  • noun
  • Division of a book
  • Ratio of something, e.g. 3 of gin, 1 of tonic
  • A line in a person's hair, made with a comb
  • Character an actor plays on stage, movie, or TV
  • Some, but not all of a specific thing
  • Role in causing something to happen
  • verb
  • To make a line in a person's hair, by using a comb
  • To move two things apart creating a space
  • To leave someone, or to break off a relationship

negative

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US / 'neɡətɪv /

UK / ˈnɛɡətɪv /

A2
  • adjecitve
  • Being harmful, unwanted or unhelpful
  • In mathematics, being less than zero
  • Focusing on the bad aspects; pessimistic
  • noun
  • The opposite to a positive electrical charge
  • In grammar, containing words such as 'no' or 'not'
  • Reply to a question or statement that means 'no'
  • Image on camera film used to make a photo

before

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US / bɪˈfɔ:(r) /

UK / bɪˈfɔr,-ˈfor /

A1
  • adverb
  • At a time earlier than the present; previously
  • conjunction
  • In advance of something happening
  • preposition
  • In front of someone or something

sine

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US / saɪn /

UK / saɪn /

C1
  • noun
  • Angle's ratio of side opposite hypotenuse

know

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US / nəʊ /

UK / noʊ /

A1
  • verb
  • To be familiar with a person or place, thing
  • To feel sure or be convinced about something
  • To discover or be aware of something
  • To be able to distinguish
  • To have knowledge of things

with

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US / wɪð /

UK / wɪð, wɪθ /

A1
  • preposition
  • As an employee of a certain company
  • Against something or someone
  • By using something as a means
  • Due to; because of
  • Including
  • Possessing as a feature
  • Related to
  • (Of the manner/attitude by which you do something)
  • Used to express that people or things are together

prime

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US / praɪm /

UK / praɪm /

A2
  • adjecitve
  • Concerning the highest in quality or value
  • Concerning the highest level of achievement
  • Most important, chief importance in value
  • verb
  • To get a person ready for; prepare someone
  • To coat with paint to prepare it for the main coat

have

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US / həv /

UK / hæv /

A1
  • verb
  • To drink, smoke, eat or use something
  • To experience the effects of something
  • To organize an event, or join in an activity
  • To own, possess, or hold something
  • To cause to happen or produce a particular effect

see

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US / si: /

UK / si /

A1
  • verb
  • To go on a date with someone
  • To predict whether something is, or will be true
  • To understand what someone says
  • To use your eyes to look at something
  • To visit a place often for sightseeing or pleasure
  • To visit or meet with someone
  • To find out by waiting or looking
  • To watch (a game, movie or TV show)

remember

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US / rɪ'membə(r) /

UK / rɪˈmɛmbɚ /

A1
  • verb
  • To give someone a gift, e.g. birthday, wedding
  • To hold a thought in your mind to not forget it
  • To bring a previous image or idea to your mind

full

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US / fʊl /

UK / fʊl /

A1
  • adjecitve
  • Containing all the parts; complete
  • Containing or holding as much as possible
  • Having had enough food
  • adverb
  • In a direct manner; completely

could

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US / kəd /

UK / kʊd /

A1
  • other
  • Past form of 'can' to mean have ability
  • Used to ask for something politely, or to offer

beauty

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US / 'bju:tɪ /

UK / ˈbjuti /

A2
  • noun
  • Something attractive that produces great pleasure
  • Physically or visually attractive person

keep

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US / ki:p /

UK / kip /

A1
  • noun
  • Large, strong tower in the middle of a castle
  • Money spent on you so you can live in a place
  • verb
  • To continue doing something; not to stop
  • To stop balls going in a goal in sports
  • To not give back or return something
  • (Of food) to not spoil or not rot
  • To take possession of something to own, e.g. a pet
  • To have and continue to hold something
  • To do something that you have promised or agreed
  • To stay in a certain place or condition
  • To maintain the rules

another

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US / əˈnʌðə(r) /

UK / əˈnʌðɚ /

A1
  • adjecitve
  • One more, but not this
  • determiner
  • One more added
  • pronoun
  • One more (thing)

over

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US / 'əʊvə(r) /

UK / ˈovɚ /

A1
  • adverb
  • Finished; no longer happening
  • Towards the ground/away from a standing position
  • noun
  • A set of six consecutive balls bowled in cricket
  • other
  • Finished
  • preposition
  • Showing the relation when a thing is above another
  • On or covering something
  • Above or across something
  • More than you need
  • From one side to the other
  • In the direction of; toward

any

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US / 'enɪ /

UK / ˈɛni /

A1
  • adjecitve
  • One, or some, no matter which
  • pronoun
  • One (thing) of many; some

put

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US / pʊt /

UK / pʊt /

A1
  • phrasal verb
  • to make a light stop shining by pressing or moving a switch
  • verb
  • To move or place a thing in a particular position
  • To rate something in importance or value
  • To write or say something in a certain manner

and

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US / ənd /

UK / ənd, ən,ænd /

A1
  • conjunction
  • Used to refer to two or more things
  • Plus; in addition; on top of that
  • Used to introduce an action that follows another

where

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US / hwɛə /

UK / hwɛr, wɛr /

A1
  • conjunction
  • At or in the place which
  • In whatever place
  • pronoun
  • Word you use to ask the location of something

from

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US / frəm /

UK / frʌm, frɑm,frəm /

A1
  • preposition
  • Starting at a particular place, time or level

far

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US / fɑ:(r) /

UK / fɑr /

A1
  • adjecitve
  • A long way away
  • Being a long way away
  • adverb
  • To a large degree or extent

back

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US / bæk /

UK / bæk /

A1
  • adjecitve
  • Farthest from the front e.g. in a classroom
  • adverb
  • Have returned to a place you were before
  • noun
  • The part of a chair that the upper body leans upon
  • Location at the rear of something
  • The reverse of side something
  • Area on the rear of your body
  • verb
  • To bet money on something
  • To go backwards in a car
  • To support and encourage someone or some cause

variable

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US / ˈveəriəbl /

UK / ˈveriəbl /

A2
  • adjecitve
  • Having the ability to change or vary
  • noun
  • Mathematical quantity with several values

drumroll

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US / 'drʌmrəʊl /

UK / ˈdrʌmˌrol /

A1
  • noun

something

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US / ˈsʌmθɪŋ /

UK / ˈsʌmθɪŋ /

A1
  • adverb
  • In some degree or measure
  • noun
  • a person or thing of some value or consequence
  • pronoun
  • Thing that is not yet known or named
  • Approximately

think

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US / θɪŋk /

UK / θɪŋk /

A1
  • verb
  • To have an idea about something without certainty
  • To have an idea, opinion or belief about something

assign

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US / ə'saɪn /

UK / əˈsaɪn /

B2
  • verb
  • To give someone a particular job to do
  • To give someone credit or blame for something

two

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US / tu: /

UK / tu /

A1
  • number
  • 2

out

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US / aʊt /

UK / aʊt /

A1
  • adverb
  • No longer stylish (e.g. fashion)
  • Away from home, business, or the usual place
  • In a direction/movement away from inside or center
  • noun
  • Act when a batter cannot continue playing
  • verb
  • To tell people a secret

come

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US / kʌm /

UK / kʌm /

A1
  • verb
  • To arrive at a place
  • (Of mail) to be delivered
  • To move toward someone; go with someone
  • To reach a certain point or state

hey

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US / heɪ /

UK / he /

A1
  • exclamation
  • What you say when you want someone's attention

call

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US / kɔ:l /

UK / kɔl /

A1
  • noun
  • A order or request for action
  • The sound an animal makes, often when in danger
  • A correct prediction, e.g. a decision in a game
  • A short visit to a place or person
  • A decision by a referee or umpire in a game
  • verb
  • To make a request or order for action
  • (Animal) to make sounds to warn of danger
  • To announce the date of an election
  • To make a judgment about something
  • To give someone or something a name
  • To guess in advance the result or outcome
  • To visit a place or person for a short time
  • To phone someone
  • To speak, announce or shout in a loud voice

minus

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US / ˈmaɪnəs /

UK / ˈmaɪnəs /

A2
  • adjecitve
  • Being close to the bottom of a letter grade
  • noun
  • Negative quality; disadvantage
  • Sign that indicates subtraction
  • preposition
  • Being less than zero in number
  • Used when subtracting one number from another
  • verb
  • To take one number away from another; subtract

leave

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US / li:v /

UK / liv /

A1
  • noun
  • Permission to do something
  • Vacation time; time off work
  • verb
  • To go away from; depart
  • To gift property to someone after you die
  • To allow something to continue happening
  • To let something remain unused or available
  • To let something or someone remain somewhere

for

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US / fə(r) /

UK / fɔr,fə /

A1
  • preposition
  • Used to show the purpose, or need of something

once

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US / wʌns /

UK / wʌns /

A1
  • adverb
  • One time; one instance
  • At a time in the past
  • conjunction
  • When; as soon as

will

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US / wɪl /

UK / wɪl /

A1
  • noun
  • Desire or choice of someone
  • Document saying who gets your money when you die
  • Strong desire to do something; determination
  • Desire to do something; strong wish to do
  • other
  • Used with verbs to express the future
  • verb
  • To leave property to someone after your death
  • To use mental effort to make something happen
  • Am (is, are) likely to do
  • To influence someone to do something
  • To make something happen by strongly wanting it

write

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US / raɪt /

UK / raɪt /

A1
  • verb
  • To compose letters and words on paper or a screen
  • To invent or create a computer program
  • To create a song or piece of music

plus

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US / plʌs /

UK / plʌs /

A2
  • noun
  • A benefit or advantage
  • preposition
  • In addition; add
  • In addition to; Indicates a number above
  • Having a value greater than zero

deserve

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US / dɪ'zɜ:v /

UK / dɪˈzɚv /

A2
  • verb
  • To be worthy of, e.g. getting praise or attention

actually

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US / ˈæktʃuəli /

UK / ˈæktʃuəli /

A1
  • adverb
  • Used to add new (often different) information

use

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US / ju:s /

UK / ju:s /

A1
  • noun
  • Ability to employ something
  • Need to enable it to work
  • Permission to do something with something
  • Act of spending money
  • Operation of a machine for its function
  • verb
  • To have; to eat, drink, consume etc.
  • To need to operate or work
  • To operate a machine, e.g. a computer
  • To treat a person badly
  • To do something with, for a task or purpose

obviously

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US / ˈɒbviəsli /

UK / ˈɑːbviəsli /

A2
  • adverb
  • In a way that is obvious/easy to see or understand
  • Subtitles section
  • Let's now see if we can use integration by parts to take the antiderivative of e to the x cosine of x dx.
  • And this one's an interesting one, and you'll see why in a few minutes.
  • Because here, if I take the derivative of either of these, it doesn't get appreciably more simple or appreciably more complicated if I were to take its antiderivative.
  • The derivative of e to the x is e to the x.
  • The antiderivative of e to the x is e to the x.
  • Derivative of cosine of x is negative sine of x.
  • Antiderivative of cosine of x is sine of x.
  • It seems to be about comparable.
  • And because of that, we can essentially, let's just arbitrarily assign one of these to be f of x and the other one to be g prime of x.
  • Neither one of them simplifies dramatically or gets more complicated when I take the derivative or the antiderivative.
  • So let's say that f of x is equal to e to the x.
  • And let's set g prime of x to be equal to cosine of x.
  • So given that, we can write over here that f of x is equal to e to the x.
  • F prime of x is equal to e to the x.
  • We can write that g prime of x, the derivative of g, is equal to cosine of x.
  • And then the antiderivative, g of x, could just be equal to sine of x.
  • So let's apply integration by parts.
  • This whole thing will simplify, too.
  • This will be equal to f of x times g of x.
  • So this is e to the x times sine of x minus the antiderivative of f prime of x.
  • Well, f prime of x is e to the x.
  • So it's going to be e to the x times g of x times sine of x.
  • So once again, this doesn't look like we've gotten, obviously, any more simple so far.
  • This antiderivative situation is about as hard as this one right over here.
  • But let's keep going to see if something interesting might happen.
  • You might guess what might happen.
  • So let's see how we can evaluate.
  • Whoops, I wanted to select that.
  • Let's think about how we can evaluate this thing here.
  • So let me copy and paste it.
  • So copy and paste.
  • So let's try to evaluate this thing right over here.
  • And similarly, we can do it using integration by parts.
  • So now let's set f of x, once again, equaling e to the x.
  • So f of x is equal to e to the x, just like we did before.
  • F prime of x is equal to e to the x.
  • We could set g prime of x.
  • G prime of x is equal to sine of x.
  • Then g of x, the antiderivative of sine of x, is negative cosine of x.
  • Now let's apply integration by parts again.
  • So this thing over here, remember, this is what we're evaluating.
  • This is what we're evaluating now.
  • We're doing a second round of integration by parts.
  • This is what we're evaluating.
  • This is equal to f of x times g of x.
  • F of x is e to the x.
  • It's e to the x.
  • Well, we have a negative out here.
  • So I'll write negative e to the x cosine of x.
  • And we got the negative from the g of x.
  • I'll put that in pink as well.
  • And then we have minus the antiderivative of f prime of x.
  • F prime of x is e to the x.
  • G of x.
  • G of x is negative cosine of x.
  • So we could put the cosine of x over here and then take the negative out of the integral.
  • And so this could become a positive.
  • It was subtracting this integral, which had a negative in it.
  • I just took the negative out.
  • But this became a positive dx.
  • So you might say, hey, Sal, it looks like we're going in circles.
  • Once again, we have another integral that was no less complicated than this one.
  • And it's actually got us back to where we were in the beginning.
  • But this is interesting.
  • What if we assigned a variable to this thing that we had in the beginning?
  • I don't know.
  • Let's call that a.
  • Let's call this thing a.
  • A is equal to this business right over here.
  • So now we can say that a is equal to this business, which is equal to e to the x sine of x.
  • Now I want to do that same pink color.
  • Sine of x.
  • And I'm just using a to simplify the notation a little bit.
  • E to the x sine of x minus this business, which we just figured out was negative e to the x cosine of x.
  • So negative e to the x cosine of x.
  • E to the x cosine of x.
  • And then this right over here is our a again.
  • A was defined as the antiderivative of e to the x cosine of x.
  • So this right over here is our a again.
  • So we could put a plus a, just like that.
  • And now let's see if we could solve for a, because a is what we cared about.
  • So let's see what we can do.
  • A is equal to e to the x sine of x, and then distribute the negative.
  • So it's going to be plus e to the x cosine of x.
  • And then minus a.
  • I'll do the a in blue.
  • Minus a.
  • I just distributed this negative sign.
  • Now if we want to solve for a, we could just add a to both sides to get all the a's on the left-hand side.
  • And then we are left with 2a is equal to this business, which I can just copy and paste.
  • 2a is equal to that.
  • So let me copy and paste that.
  • 2a is equal to that.
  • Or a, which is what we cared about in the beginning, a is equal to, and now we definitely deserve a drumroll, a is equal to all of this business divided by 2. We're just solving for a.
  • Is equal to all of that business divided by 2. And remember, a is just what we wanted to solve in the beginning.
  • A is equal to the antiderivative of e to the x cosine of x dx.
  • And we're done.
  • We were able to use integration by parts twice to come around full circle to get back to our original expression.
  • But then we solved for that original expression.
  • We had an equation in terms of that original expression.
  • I just set a to be equal to it, so I didn't have to rewrite it over and over again.
  • But we have our original expression, the antiderivative of e to the x cosine of x is equal to all of this business, which is kind of a neat expression right here.
  • It has a kind of a beauty to it.
  • It has an e to the x.
  • It's got sine of x plus cosine of x.
  • You're taking somehow the average of these two things.
  • It seems kind of just like a neat thing.