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Integration by parts

Hi, I was trying this question from madas maths, but I'm not sure how to do it as whichever term I put as u and dv/dx, I always end up having to integrate something of the style e^x(trig(x))
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tsr integral.png
tsr integral.png6.5KB
Original post by Cuspake
tsr integral.png

If you do integration by parts twice ...
This integral is pretty interesting and classic. I'll start you off by letting I="the integral".
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Original post by tonyiptony
This integral is pretty interesting and classic. I'll start you off by letting I="the integral".

As in letting 1 = dv/dx?
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Original post by mqb2766
If you do integration by parts twice ...

What should I put as u and dv/dx please?
Original post by Cuspake
What should I put as u and dv/dx please?

One is the exponetial, the other is the trig term.
Original post by Cuspake
As in letting 1 = dv/dx?

As @mqb2766 suggests, one is the trig, the other is the exponential. It really doesn't matter either case.
Now if you let I="the integral", something interesting happens if you IBP twice. Try it!
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Do I use 1 as the integral both times round?

Original post by tonyiptony
As @mqb2766 suggests, one is the trig, the other is the exponential. It really doesn't matter either case.
Now if you let I="the integral", something interesting happens if you IBP twice. Try it!
In cases where you get back your intial integral just replace it with I, that way you can get rid of it since I would always be your initial integral
(edited 1 year ago)
2022-05-12 (2).png454.7KB
Original post by orion reid
...

Best to let the OP have time to work through their problems, as per the forum guidelines. Can you delete pls.
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Original post by orion reid
2022-05-12 (2).png
I have attached a picture of a worked solution.
In cases where you get back your intial integral just replace it with I, that way you can get rid of it since I would always be your initial integral

Ohhh, thank you, that makes a lot of sense, thank you. I now understand what replacing the integral with I actually means aha
Original post by mqb2766
Best to let the OP have time to work through their problems, as per the forum guidelines. Can you delete pls.

Aye aye

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