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Integration by 3 parts

How on earth can I integrate this:

x(ex)(sin(x))-x*(e^x)(sin(x))

?
Reply 1
first find exsinx dx\int e^xsinx\ dx then do IBP on xexsinx dx\int -xe^xsinx\ dx with u=xu=-x and dv/dx=exsinxdv/dx = e^xsinx
Reply 2
Original post by Pheylan
first find exsinx dx\int e^xsinx\ dx then do IBP on xexsinx dx\int -xe^xsinx\ dx with u=xu=-x and dv/dx=exsinxdv/dx = e^xsinx


I got 12ex(sinxcosx)\frac{1}{2}e^x(sinx - cosx) for first bit

Is this anywhere near right?
Reply 3
Original post by Dagnabbit
I got 12ex(sinxcosx)\frac{1}{2}e^x(sinx - cosx) for first bit

Is this anywhere near right?


if memory serves, yes
Reply 4
Original post by lilangel890
Do you have to do it by parts? If not you can do it by log differentiation and it's much simpler:

Let y=xexsinxy=-xe^xsinx

Log both sides. We know ln(abc) = lna + lnb + lnc. Use this to split the expression and then differentiate wrt x (remembering left hand side is implicit, so you get 1/y dy/dx)

Once you multiply up by the y and put in a nicer form you will get the required answer


gosh, that's so much simpler. Cheers!
Reply 5
Original post by Dagnabbit
gosh, that's so much simpler. Cheers!


it's also incorrect
:mad:
Original post by Pheylan
it's also incorrect


Get your facts straight nub. I did it and checked it in Matlab and it's right :wink:
Reply 7
Original post by lilangel890
:mad:

Get your facts straight nub. I did it and checked it in Matlab and it's right :wink:


what answer did you get?
Original post by lilangel890
:mad:

Get your facts straight nub. I did it and checked it in Matlab and it's right :wink:

What on Earth are you on about? You've just differentiated it, whereas the OP is trying to integrate... Pheylan's advice is correct.
Reply 9
Original post by lilangel890
:mad:

Get your facts straight nub. I did it and checked it in Matlab and it's right :wink:


Yes but you've invented your own question.
Reply 10
While on integration

integral of

e^(-x) * sine sqared x
???
Original post by ck_no_1
While on integration

integral of

e^(-x) * sine sqared x
???


It's a tricky one. You need to use IBP 3 times and also be able to spot the trick for integrating things of the form ekxsin(nx)e^{kx}\sin (nx). The answer isn't that messy.
Reply 12
for the complex conscious... or if you just hate using paper :smile:
Unparseable latex formula:

-\int xe^{(1+i)x}\mbox{d}x=\frac{e^{(i+1)x}\left[1-\left(i+1\right)x\right]}{\left(1+i\right)^{2}}+C= \frac{e^{x}}{2}\left(ie^{ix}(1+i)x-ie^{ix}\right)+C



taking the imaginary part gives: ex2[(cosxsinx)xcosx]+C\frac{e^{x}}{2}\left[\left(\cos x-\sin x\right)x-\cos x\right]+C and we're done.

all you need to know is that eix=cosx+isinx e^{ix}= \cos x+i \sin x and treat i as a constant when integrating

For the integral
Unparseable latex formula:

\int e^{-x}\sin^{2}x\mbox{d}x

, try the identity sin2x=12(1cos2x) \sin^{2}x=\frac{1}{2}\left(1-\cos2x\right) and integrate by parts as normally

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