I think I set up this problem right, but now I'm just left with another integral on the end that I would have to use integration by parts again with. Is this right? What's the best way to solve the initial problem?
#Integration by parts help
edgy citrus
mystic pollenBOT
neat meadow
Maybe you should try to use the fact that sin(5x) can be rewrite im(e^i5x) and after you have to integrate something more gentle but two integrations by part work also
teal forge
For integrals like
$I = \int e^x\sin{x},dx$,
integrationg by parts does yield a new integral that can also be integrated by parts. So long as the same function is chosen for integration each time, applying parts twice yields results like
$I = f(x) + g(x) - I$
which can then be rearrange to
$I = \frac{1}{2}\left[f(x) + g(x)\right] + C$, for some constant $C$.
gaunt coveBOT
Jay
teal forge
$I = \int e^x\sin{x}, dx = \int \sin{x}, de^x$
$= e^x\sin{x} - \int e^x, d\sin{x} = e^x\sin{x} - \int e^x\cos{x}, dx$
$= e^x\sin{x} - \int \cos{x}, de^x = e^x\sin{x} - e^x\cos{x} + \int e^x, d\cos{x}$
$= e^x\sin{x} - e^x\cos{x} + \int e^x\left(-\sin{x}\right), dx = e^x\left(\sin{x} - \cos{x}\right) - \int e^x\sin{x}, dx$
$\implies I = e^x\left(\sin{x} - \cos{x}\right) - I$
$\implies I = \frac{e^x}{2}\left(\sin{x} - \cos{x}\right) + C \qquad \text{for some constant}\ C$
gaunt coveBOT
Jay
