Here's the question you clicked on:
jennychan12
integral from 0 to 1 of [e^x(cos(e^x))]dx ?
\[\int\limits_{0}^{1} [e^x \cos (e^x)]dx\]
Do a u-substitution with \(u=\sin(e^x)\), and your solution should pop right out.
but if you do that, then du = e^xcos(e^x) and there's no cos (e^x) in the question.
that'd just be ucosu
\(u=\sin(e^x)\implies du=e^x\cos(e^x)dx\). So your integral becomes\[\int du=u\]
Substitute back for \(u\), and you get \[\int_0^1 e^x\cos(e^x)dx=\sin(e^x)|_0^1\]
u =e^x du = e^x dx \(\int cos udu\) ohh..now i could say that u =sin e^x is a better substitution..
oh wait. my bad. i thought u said u = sin u ok i see now
Yup, I like to call my method "guessing the solution and proving you're right"
Although if you really had no clue of the solution, \(u=e^x\) would be a fine substitution. You would just have to integrate by parts.
whenever i see a non-standard angle with sin/cos/.. i'll call that as 'bad angle' and put u= bad angle.... example : sin x^2 , cos log x ....
how do we need integration by parts ?? O.o u= e^x du=e^xdx
Oh. Right. You don't. Ignore my ramblings.