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AravindG
 2 years ago
integral (e^x(xcos x+sin x))dx=?
AravindG
 2 years ago
integral (e^x(xcos x+sin x))dx=?

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AravindG
 2 years ago
Best ResponseYou've already chosen the best response.1\[ \large \int\limits e^x(xcos x+\sin x)dx\]

AravindG
 2 years ago
Best ResponseYou've already chosen the best response.1I have been stuck on this question for sometime..I am missing something somewhere.

AravindG
 2 years ago
Best ResponseYou've already chosen the best response.1@ganeshie8 , @jim_thompson5910 Can you help?

anonymous
 2 years ago
Best ResponseYou've already chosen the best response.0its a case in integration of by parts!!

anonymous
 2 years ago
Best ResponseYou've already chosen the best response.0\[=\int\limits (x \cos x) e ^{x} dx+\int\limits (\sin x) e ^{x}dx\] \[=x \cos x e ^{x}\int\limits \left\{ x \left( \sin x \right)+\cos x *1 \right\}e ^{x} dx+\[\sin x e ^{x}\int\limits \cos x e ^{x}dx\]+c\] there is some mistake,try to find out.

AravindG
 2 years ago
Best ResponseYou've already chosen the best response.1I am thinking of doing it in the form e^x(f(x)+f'(x)) By supplying terms and subtracting.

anonymous
 2 years ago
Best ResponseYou've already chosen the best response.0i have tried that also.

AravindG
 2 years ago
Best ResponseYou've already chosen the best response.1@surjithayer How did you get integral (xcos x)e^x dx=(xcos x)(e^x)

anonymous
 2 years ago
Best ResponseYou've already chosen the best response.0you are doing that right and its integration is e^x.f(x)+c

AravindG
 2 years ago
Best ResponseYou've already chosen the best response.1@divu.mkr can you explain?

anonymous
 2 years ago
Best ResponseYou've already chosen the best response.0everything i wrote is gone

anonymous
 2 years ago
Best ResponseYou've already chosen the best response.0it all ends up being in terms of:\[\int\limits_{}^{}e^x cosx dx= \frac{ 1 }{ 2 }e^x(sinx + cosx) + C\] and\[\int\limits_{}^{}e^x sinx dx= \frac{ 1 }{ 2 }e^x(sinx  cosx) + C\]. a rather simple integration by parts. and seperating it like surji suggested and integrating only the first term by parts one gives:\[\frac{ x }{ 2 }e^x(sinx+cosx)  \frac{ 1 }{ 2 }\int\limits_{}^{} e^x (sinx + cosx)dx + C + \int\limits_{}^{}e^x sinx dx\]

anonymous
 2 years ago
Best ResponseYou've already chosen the best response.0was my own fault. i pressed a wrong key and then backspace

anonymous
 2 years ago
Best ResponseYou've already chosen the best response.0the integral of that, by parts once, gives the first two terms of the last line. taking u = x dv = e^xcosx

anonymous
 2 years ago
Best ResponseYou've already chosen the best response.0i wouldn't mind solving it entirely if you're not convinced :)

AravindG
 2 years ago
Best ResponseYou've already chosen the best response.1Its just I want to to do this qn in form e^x((fx)+f'(x)) Format and not apply parts.

anonymous
 2 years ago
Best ResponseYou've already chosen the best response.0i'm not familiar with that method at all. only parts

anonymous
 2 years ago
Best ResponseYou've already chosen the best response.0did you want to see parts or you know how?

AravindG
 2 years ago
Best ResponseYou've already chosen the best response.1yeah I want to see parts.

anonymous
 2 years ago
Best ResponseYou've already chosen the best response.0i guess the only part is the:\[\int\limits_{}^{} e^x cosx dx\] u = e^x; du = e^x dx dv = cosx dx; v = sinx \[\int\limits_{}^{} e^x cosx dx = e^xsinx  \int\limits_{}^{} e^x sinx dx\] u = e^x; du = e^x dx dv = sinx dx; v = cosx \[\int\limits_{}^{}e^x cosx dx= e^x sinx  \left( e^x cosx + \int\limits_{}^{}e^x cosxdx \right)\] \[\int\limits_{}^{}e^x cosx dx = e^x(cosx + sinx)  \int\limits_{}^{}e^x cosx dx\]\[2\int\limits_{}^{}e^x cosx dx = e^x(sinx + cosx)\]\[\int\limits_{}^{}e^x cosx dx = \frac{ 1 }{ 2 }e^x (sinx + cosx)\]

anonymous
 2 years ago
Best ResponseYou've already chosen the best response.0do you have the answer..? i got one :D

AravindG
 2 years ago
Best ResponseYou've already chosen the best response.1yes final answer is 1/2e^x(xsin x+xcos xcos x)+c

AravindG
 2 years ago
Best ResponseYou've already chosen the best response.1@Euler271 Yeah I get it now thanks!

anonymous
 2 years ago
Best ResponseYou've already chosen the best response.0i have tried that also.

ybarrap
 2 years ago
Best ResponseYou've already chosen the best response.0Lots of integration by parts. Work on this in pieces (I've reused u and v several times but hopefully you can tell which is which by context): $$ \int e^x(x\cos x+\sin x)dx\\ \int xe^x \cos x ~dx+\int xe^x \sin x~dx\\ \text{For }\int xe^x \cos x ~dx\\ \text{Let }u=x,dv=e^x\cos x\\du=x,v=\int e^x\cos x dx\\ \text{For }\int e^x\cos x dx\\ \text{Let }u=e^x,dv=\cos x\\ du=e^x,v=\sin x\\ \text{Then }uv\int vdu=\\ e^x\sin x\int e^x\sin x dx\\ \int e^x\sin x dx=\\ \text{Let }u=e^x,dv=\sin x\\ \text{then }du=e^x,v=\cos x\\ uv\int vdu = \\ e^x\cos x+\int e^x\cos x dx\\ \text{Combining these results:}\\ \int e^x\cos x=\cfrac{e^x}{2}(\sin x+\cos x)\\ \text {So we now have }\\ \int xe^x \cos x ~dx\\ =\cfrac{xe^x}{2}(\sin x + \cos x)\int \cfrac{e^x}{2}(\sin x + \cos x)dx+\\ \qquad \cfrac{e^x}{2}(\sin x  \cos x)\\ \text{Take, }\int\frac{e^x}{2}(\sin x + \cos x)dx\\ \int\frac{e^x}{2}\sin x + \int \cfrac{e^x}{2}\cos xdx\\ =\cfrac{e^x}{4}(\sin x cos x)+\cfrac{e^x}{4}(\sin x +cos x)\\ =\cfrac{e^x}{2}\sin x\\ \text{So, }\\ \int xe^x \cos x ~dx\\ =\cfrac{xe^x}{2}(\sin x + \cos x)\cfrac{e^x}{2}\sin x+\cfrac{e^x}{2}(\sin x  \cos x)\\ =\cfrac{e^x}{2}(x\sin x + (x1)\cos x)\\ $$ Whew!! That's it!
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