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