How do you find the indefinite integral of int e^(2x)sin(7x)dxe2xsin(7x)dx?

1 Answer
Sep 20, 2015

1/53e^(2x)(2sin 7x-7cos 7x)+C153e2x(2sin7x7cos7x)+C

Explanation:

Integration by parts:

intudv=uv-intvduudv=uvvdu

I=inte^(2x)sin 7xdxI=e2xsin7xdx

e^(2x)=u => 2e^(2x)dx=due2x=u2e2xdx=du
dv=sin 7xdx => v=intsin 7xdx=-1/7cos 7xdv=sin7xdxv=sin7xdx=17cos7x

I=-1/7e^(2x)cos 7x+2/7inte^(2x)cos 7xdxI=17e2xcos7x+27e2xcos7xdx

Again:

e^(2x)=u => 2e^(2x)dx=due2x=u2e2xdx=du

dv=cos 7xdx => v=intcos 7xdx=1/7sin 7xdv=cos7xdxv=cos7xdx=17sin7x

I=-1/7e^(2x)cos 7x+2/7[1/7e^(2x)sin7x-2/7inte^(2x)sin7xdx]I=17e2xcos7x+27[17e2xsin7x27e2xsin7xdx]

I=-1/7e^(2x)cos 7x+2/49e^(2x)sin7x-4/49inte^(2x)sin7xdxI=17e2xcos7x+249e2xsin7x449e2xsin7xdx

I=-1/7e^(2x)cos 7x+2/49e^(2x)sin7x-4/49II=17e2xcos7x+249e2xsin7x449I

I+4/49I=-1/7e^(2x)cos 7x+2/49e^(2x)sin7xI+449I=17e2xcos7x+249e2xsin7x

53/49I=-1/7e^(2x)cos 7x+2/49e^(2x)sin7x5349I=17e2xcos7x+249e2xsin7x

I=-7/53e^(2x)cos 7x+2/53e^(2x)sin7x+CI=753e2xcos7x+253e2xsin7x+C

I=1/53e^(2x)(2sin 7x-7cos 7x)+CI=153e2x(2sin7x7cos7x)+C