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

1 Answer
Nov 7, 2016

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.