Question

How do you integrate `x * cos^2 (x)`?

Answer 1

`x^2/4+(x/4)sin2x+cos(2x)/8+c`.where c is integration constant .

Explanation:

`int (xcos^2x)dx`
`=(1/2)int(x2cos^2x)dx`
now `int(x2cos^2x)dx`
=`int{x(1+cos2x)}dx`
=`intxdx+int(xcos2x)dx`
[integration by parts]
=`x^2/2+x(int(cos2x)dx)-int[(dx/dx)int(cos2x)dx]dx`
=`x^2/2+(xsin(2x))/2` `-int((sin2x)/2)dx`
=`x^2/2+(xsin(2x))/2` `+(cos2x)/4` `+c`
hence the value of the integral :`x^2/4+(xsin(2x))/4+cos(2x)/8+c`,where c is constant of integration.