How do you integrate xsin2x by parts from [0,π]?

1 Answer
Feb 5, 2017

π0xsin2xdx=π2

Explanation:

Consider the integral:

π0xsin2xdx

Note that:

d(12cos2x)=sin2xdx

so we can integrate by parts:

π0xsin2xdx=π0xd(12cos2x)

π0xsin2xdx=[xcos2x2]π0+12π0cos2xdx

π0xsin2xdx=[xcos2x2]π0+14[sin2x]π0

π0xsin2xdx=[πcos(2π)2+0]+[00]

π0xsin2xdx=π2