Integrate ⌠( xcos^2(x))dx
We must first use trigonometric identities to simplify cos2(x). We can use the formula cos(A+B) = cos(A)cos(B) - sin(A)sin(B) , where A=x and B=x, so that we get cos(2x) = cos2(x) - sin2(x) = 2cos2(x) - 1. Rearranging this we find that cos2(x) = 1/2 + (cos(2x))/2 This gives us (⌠(x+xcos(2x))dx)/2 = 1/2(⌠( x)dx + ⌠( xcos(2x))dx) = x2 /4 + ⌠( xcos(2x))dx) /2 We can then use the integration by parts formula, ⌠(udv)dx = uv - f(vdu)dx , where u=x and dv=cos(2x), so that we get x2 /4 + ⌠( xcos(2x))dx) /2 = x2 /4 + xsin(2x)/4 - ⌠(sin(2x))dx)/4 = x2 /4 + xsin(2x)/4 + cos(2x)/8 Hence,the final answer is x2 /4 + xsin(2x)/4 + cos(2x)/8 + c
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