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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