Posted by **chris** on Sunday, February 27, 2011 at 2:15pm.

integration of x^2 * cos^2(x)

- math -
**agrin04**, Monday, February 28, 2011 at 1:05am
|x^2*cos^2(x) dx =

=|x^2*(1/2)(1+cos(2x)) dx

=(1/2)|x^2 dx + (1/2)|x^2*cos(2x) dx

=(1/6)x^3 + (1/2)|x^2*cos(2x) dx

Using integration by part:

u = x^2

du = 2x dx

dv = cos(2x) dx

v = (1/2) sin(2x)

|x^2*cos(2x) dx =

= (1/2)x^2*sin(2x) - |xsin(2x) dx

Again, using integration by part:

u = x

du = dx

dv = sin(2x) dx

v = -(1/2)cos(2x)

|x^2*cos(2x) dx =

= (1/2)x^2*sin(2x) - {-(1/2)x*cos(2x) + (1/2)|cos(2x) dx}

= (1/2)x^2*sin(2x) + (x/2)cos(2x) - (1/4)sin(2x) + const

Finally, we have:

|x^2*cos^2(x) dx =

= (1/6)x^3 + (1/4)x^2*sin(2x) + (x/4)cos(2x) - (1/8)sin(2x) + const

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