integral of xsin(2x)dx?

Question

im not sure how you get a cos out of this integral

im not sure how you get a cos out of this integral so could you show me multiple steps please? thank you.

Answer 1

Let U = -1/2 cos(2x)
V=x
dU = sin(2x) dx
dV=dx
Since int (V . dU) = V U - int (U .dV)
So int ( xsin(2x)dx) = -1/2 x cos(2x) - int (-1/2 cos (2x) dx)
= -1/2 x cos(2x) + 1/4 sin(2x) + C (integration constant)

Answer 2

Integrate by parts:

(sin(2x) - 2x cos (2x))/4

Answer 3

I = ∫ x.sin 2x dx
Use integration by parts:-
∫u .(dv/dx).dx = u.v - ∫ v.(du/dx).dx
u = x and dv/dx = sin 2x
du/dx = 1 and v = - cos 2x / 2
I = - x.cos 2x / 2 + (1/2) ∫ cos 2x.dx
I = - x.cos 2x / 2 + (1/4).sin 2x + C