Integration by parts

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

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.

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

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)

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

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!)

ofc thank you!!

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

ahhhh okay i’ll try that thanks!

Using the double angle identity is key here. You need to group something with $e^{\cos x}$ 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).

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

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

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.

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

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

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.