# Integration by parts

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Hi!

I’m not sure how you would go about this question and the mark scheme isn’t any help.

If you use LATE you should take sin2x as ‘u’ and then integrate e^cosx but apparently that’s not possible to integrate? The final answer is 2.

Thanks

I’m not sure how you would go about this question and the mark scheme isn’t any help.

If you use LATE you should take sin2x as ‘u’ and then integrate e^cosx but apparently that’s not possible to integrate? The final answer is 2.

Thanks

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#2

(Original post by

Hi!

I’m not sure how you would go about this question and the mark scheme isn’t any help.

If you use LATE you should take sin2x as ‘u’ and then integrate e^cosx but apparently that’s not possible to integrate? The final answer is 2.

Thanks

**Piza**)Hi!

I’m not sure how you would go about this question and the mark scheme isn’t any help.

If you use LATE you should take sin2x as ‘u’ and then integrate e^cosx but apparently that’s not possible to integrate? The final answer is 2.

Thanks

P.S. I would use double angle identity here.

Last edited by RDKGames; 4 days ago

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(Original post by

It’s not possible so you might try the setup the other way for u and v’

P.S. I would use double angle identity here.

**RDKGames**)It’s not possible so you might try the setup the other way for u and v’

P.S. I would use double angle identity here.

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#4

You can also use the DI method, it is a very quick, efficient way of doing integration by parts

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#5

(Original post by

You can also use the DI method, it is a very quick, efficient way of doing integration by parts

**Joshwoods01**)You can also use the DI method, it is a very quick, efficient way of doing integration by parts

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#6

I'll use this as an example

choose something to integrate and differentiate and create a table like so including the plus and minus signs

keep differentiating and integrating until either:

1. the differential is zero

2.you can easily integrate the product of the horizontal

3. You arrive at an almost identical point as the question (like my example, the derivative has e^2x and the integral has sin(x)

take the product of all the diagonal and add them together

take the integral of the product of the horizontal, you will end up having a term that is equal to the original question but is scaled by some value, bring it all to one side and divide by that scalar, you have your answer, then plug in your bounds, with your question, differentiate e^cos x and integrate sin(2x)

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(Original post by

I'll use this as an example

choose something to integrate and differentiate and create a table like so including the plus and minus signs

keep differentiating and integrating until either:

1. the differential is zero

2.you can easily integrate the product of the horizontal

3. You arrive at an almost identical point as the question (like my example, the derivative has e^2x and the integral has sin(x)

take the product of all the diagonal and add them together

take the integral of the product of the horizontal, you will end up having a term that is equal to the original question but is scaled by some value, bring it all to one side and divide by that scalar, you have your answer, then plug in your bounds, with your question, differentiate e^cos x and integrate sin(2x)

**Joshwoods01**)I'll use this as an example

choose something to integrate and differentiate and create a table like so including the plus and minus signs

keep differentiating and integrating until either:

1. the differential is zero

2.you can easily integrate the product of the horizontal

3. You arrive at an almost identical point as the question (like my example, the derivative has e^2x and the integral has sin(x)

take the product of all the diagonal and add them together

take the integral of the product of the horizontal, you will end up having a term that is equal to the original question but is scaled by some value, bring it all to one side and divide by that scalar, you have your answer, then plug in your bounds, with your question, differentiate e^cos x and integrate sin(2x)

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#8

(Original post by

wowww never seen that method before, thanks I’ll try it out!

**Piza**)wowww never seen that method before, thanks I’ll try it out!

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(Original post by

It may look quite long winded but once you start to use it, it becomes so quick! Can save you a good few minutes in the exam (just mention that that is the method you're using!)

**Joshwoods01**)It may look quite long winded but once you start to use it, it becomes so quick! Can save you a good few minutes in the exam (just mention that that is the method you're using!)

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#10

(Original post by

ofc thank you!!

**Piza**)ofc thank you!!

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#11

**Joshwoods01**)

You can also use the DI method, it is a very quick, efficient way of doing integration by parts

The key observation here is:

(Original post by

ahhhh okay i’ll try that thanks!

**Piza**)ahhhh okay i’ll try that thanks!

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#12

(Original post by

Using the double angle identity is key here. You need to group something with in order to get something you can integrate by recognition (or by substitution, but if you can't recognize it, you probably won't be able to spot what to do).

**DFranklin**)Using the double angle identity is key here. You need to group something with in order to get something you can integrate by recognition (or by substitution, but if you can't recognize it, you probably won't be able to spot what to do).

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#13

**Piza**)

Hi!

I’m not sure how you would go about this question and the mark scheme isn’t any help.

If you use LATE you should take sin2x as ‘u’ and then integrate e^cosx but apparently that’s not possible to integrate? The final answer is 2.

Thanks

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#14

(Original post by

Is this a real exam question or somebody's made-up question? Any "normal" person would probably replace sin2x with 2sinxcosx and then sub u = cos x, leaving a straightforward application of IBP to the 'u'-integral, rather than trying to do IBP on the initial integral

**davros**)Is this a real exam question or somebody's made-up question? Any "normal" person would probably replace sin2x with 2sinxcosx and then sub u = cos x, leaving a straightforward application of IBP to the 'u'-integral, rather than trying to do IBP on the initial integral

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sorry the reply took so long, but I ended up using both integration by parts and substitution because I kept ending up at dead ends by just doing it with parts. Here’s my working I subbed in the values and got the correct answer of two. (there should be an extra ‘u’ i just missed it out in a line.

Haven’t tried the DI method yet but i’m sure I will before exams haha

Haven’t tried the DI method yet but i’m sure I will before exams haha

Last edited by Piza; 3 days ago

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(Original post by

It looks like a question from Madasmaths (created by the TSR user TeeEm). Except for substitution, exam questions rarely force you to use a certain integration method.

**Notnek**)It looks like a question from Madasmaths (created by the TSR user TeeEm). Except for substitution, exam questions rarely force you to use a certain integration method.

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#17

**davros**)

Is this a real exam question or somebody's made-up question? Any "normal" person would probably replace sin2x with 2sinxcosx and then sub u = cos x, leaving a straightforward application of IBP to the 'u'-integral, rather than trying to do IBP on the initial integral

[I was looking at this as analogous to integrating x^3 exp(-x^2) by parts and would have done it directly ].

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#18

(Original post by

sorry the reply took so long, but I ended up using both integration by parts and substitution because I kept ending up at dead ends by just doing it with parts. Here’s my working I subbed in the values and got the correct answer of two. (there should be an extra ‘u’ i just missed it out in a line.

Haven’t tried the DI method yet but i’m sure I will before exams haha

**Piza**)sorry the reply took so long, but I ended up using both integration by parts and substitution because I kept ending up at dead ends by just doing it with parts. Here’s my working I subbed in the values and got the correct answer of two. (there should be an extra ‘u’ i just missed it out in a line.

Haven’t tried the DI method yet but i’m sure I will before exams haha

**definite**integral - you can convert the x limits to u-limits so you can put these in directly when you have the u-integral evaluated and then you never need to write out the full version in terms of x at the end

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#19

(Original post by

Obviously I'm not "normal"...

[I was looking at this as analogous to integrating x^3 exp(-x^2) by parts and would have done it directly ].

**DFranklin**)Obviously I'm not "normal"...

[I was looking at this as analogous to integrating x^3 exp(-x^2) by parts and would have done it directly ].

Mind you, I have no idea what "LATE" and "DI" methods are supposed to be - there seems to be a modern tendency to pile methods on top of methods for the sake of having an acronym, instead of just following the basic principle of IBP. Apart from slightly more subtle examples where you use "1" as one of the functions, there are usually only 2 choices for what to differentiate vs what to integrate, and you pretty soon find out if you've made the wrong one

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#20

(Original post by

Mind you, I have no idea what "LATE" and "DI" methods are supposed to be - there seems to be a modern tendency to pile methods on top of methods for the sake of having an acronym, instead of just following the basic principle of IBP.

**davros**)Mind you, I have no idea what "LATE" and "DI" methods are supposed to be - there seems to be a modern tendency to pile methods on top of methods for the sake of having an acronym, instead of just following the basic principle of IBP.

Although like you I've never used them, so I may be talking nonsense!

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