Question #d6918

1 Answer
Nov 25, 2017

e^4(-1/2xcos(2x)+1/4sin(2x))+C

Explanation:

inte(e^3xsin(2x)dx)=inte^4xsin(2x)dx

e^4 is a constant and can be pulled out:

=e^4intxsin(2x)dx

Let's use integration by parts since we have a product of two functions:

u=x and dv=sin(2x)dx. Then:

du=dx and v=-1/2cos(2x)

intudv=uv-intvdu

intxsin(2x)dx=-1/2xcos(2x)-int-1/2cos(2x)dx=-1/2xcos(2x)+1/2intcos(2x)dx

=-1/2xcos(2x)+1/2*1/2sin(2x)+C

Now we need to make sure we do not forget about the e^4:

e^4intxsin(2x)dx=e^4(-1/2xcos(2x)+1/4sin(2x))+C