Integrate: xsin(6x)dx. I can't seem to be able to figure out what u-substitution I should use.?

Answer 1

xsin(6x)dx.
Let u = 6x so du= 6/dx , dx = du/6, x = u/6
Thus u/6 sinu du/6 = (1/36) usinu du
integral = 1/36(sinu - ucosu) +C
= (1/36) (sin6x -6xcos 6x) + C

Answer 2

Integration by parts.
u = one that differentiates to give an easier integral

u = x
dv = sin(6x) dx

du = dx
v = -1/6 cos(6x)

integral = -x/6 * cos(6x) + 1/6 * int cos(6x) dx
= -x/6 * cos(6x) + 1/36 sin(6x) + constant

Answer 3

Mental substitution:
∫xsin(6x)dx
= (1/36)∫6xsin(6x)d6x
= (1/36)∫-6xdcos(6x)
= (1/36)[-6xcos(6x) + ∫cos(6x)d6x]
= (1/36)[-6xcos(6x) + sin(6x)]