maths integration by parts

Question

using integration by parts integrate
1)  ∫ xsin2x dx
2)  ∫ xarctanx dx
3) lnx/(1+x)² dx
^{4) ∫ (lnx)2} dx

Answer 1

1) let u=x so du = dx; let dv = sin 2x dx so v = -½cos 2x.

∫ xsin2x dx = uv - ∫vdu = -½x cos 2x - ∫ (-½ cos 2x) dx.

Now you do antidifferentiate: ∫ (-½ cos 2x) dx = - ¼ sin 2x and this substitutes into the above, giving

∫ xsin2x dx = -½x cos 2x - (- ¼ sin 2x) = -½x cos 2x + ¼ sin 2x + C.

The TYPING is harder than the Calculus. I suggest that you check this result by differentiating it; you should get the original integrand back.