Question

\[Determine: \int\limits_{}^{}e^xcosxdx\] Through repeated partial integration.

Answer 1

use integration by parts, u is cos(x) and v is e^x

Answer 2

http://answers.yahoo.com/question/index?qid=20080413170149AA24nPx

Answer 3

\[u = e^x ~~and~~ dv = cosx\]
\[du = e^x dx~~ and~~ v = sinx\]

\[∫u dv = uv - ∫v du\]
\[= e^x sinx - ∫e^x sinx dx\]

integration by parts again,
\[u = e^x ~~and~~ dv = sin~x\]
\[du = e^x dx~~ and ~~v = -cosx\]

\[= e^x sinx - [-e^x cosx - ∫-e^x cosx dx]\]
\[= e^x sinx - [-e^x cosx + ∫e^x cosx dx]\]
\[= e^x sinx + e^x cosx - ∫e^x cosx dx\]

if you rewrite it like this,
\[∫e^x cosx dx = e^x sinx + e^x cosx - ∫e^x cosx dx\]

you can see that that there are\[ ∫e^x cosx dx\] on both side, so, just add \[∫e^x cosx dx\] to both sides
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so you got
\[2∫e^x cosx = e^x sinx + e^x cosx\]
\[∫e^x cosx = \frac{1}{2}(e^x sinx + e^x cosx) +constant\]

Answer 4

A bit more general approach
http://answers.yahoo.com/question/index?qid=20080723161756AA9xWm6

Answer 5

Thanks! The cycling really circulated my mind, haha :s Glad it had a relatively easy solve. :)