Question

How do you integrate `int xsin(2x)` by integration by parts method?

Answer 1

`=1/4sin(2x) - x/2cos(2x) + C`

Explanation:

For `u(x), v(x)`

`int uv'dx = uv ' - int u'vdx`

`u(x) = x implies u'(x) = 1`

`v'(x) = sin(2x) implies v(x) = -1/2cos(2x)`

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

`= -x/2cos(2x)+1/4sin(2x)+C`