e^x sin x dx?

Question

Integration by parts - Step by step

Answer 1

Int (udv)=uv-Int(vdu)
Let
u=sin(x) ; du=cos(x)dx
dv=e^xdx ; v=e^x
Then
=e^xsin(x)-Int(e^xcos(x)dx)

Well, Int(e^xcos(x)dx) can be integrated by parts also
Let
u=cos(x) ; du=-sin(x)dx
dv=e^xdx ; v=e^x
Then
=e^xcos(x) + Int(e^xsin(x)dx)

Int(e^x*sinxdx) = e^xsin(x) - [e^xcos(x)+Int(e^xsin(x)dx)]
2Int(e^x*sinxdx) = e^x(sin(x) - cos(x))
So
Int(e^x*sinxdx) = [e^x(sin(x) - cos(x))]/2 + c

Answer 2

You do it