What is the integration of xsin2x dx?

Usually I like to write/explain it in details...

It's integration by Parts:

dv = sin(2x) => v = -1/2 cos(2x)

u = x => du = dx

∫ u dv = uv - ∫ v du

∫ x∙sin(2x) dx = (x) * (-1/2) cos(2x) - ∫ (-1/2)cos(2x) dx

∫ x∙sin(2x) dx = (-1/2) * x cos(2x) + ∫ (1/2)cos(2x) dx

∫ x∙sin(2x) dx = - 1/2 * x∙cos(2x) + 1/4 * sin(2x)

I = ∫ u (dv/dx) dx = uv - ∫ v (du/dx) dx

u = x and dv/dx = sin 2x
du/dx = 1 , v = (-1/2) cos 2x

I = (-1/2) (x) cos 2x + (1/2) ∫ cos 2x dx
I = (-1/2) (x) cos 2x + (1/4) sin 2x + C

buy a calculator. and read your textbook. and go to tutorials. ;D

Integration by Parts:
∫ x∙sin(2x) dx = -½ x∙cos(2x) + ½ ∫ cos(2x) dx
∫ x∙sin(2x) dx = -½ x∙cos(2x) + ¼ sin(2x)