Integral X Cos X Dx

Mathmom

Yes, you need to use integration by parts twice.

∫ x² cos(x) dx

u = x² . . . . . . . dv = cos(x) dx

du = 2x dx . . . v = sin(x)

∫ u dv = u*v – ∫ v du

∫ x² cos(x) dx = x² sin(x) – ∫ 2x sin(x) dx

∫ x² cos(x) dx = x² sin(x) – 2 ∫ x sin(x) dx

Integrate by parts again:

u = x . . . . . dv = sin(x) dx

du = dx . . . v = -cos(x)

∫ x² cos(x) dx = x² sin(x) – 2 (-x cos(x) – ∫ – cos(x) dx)

∫ x² cos(x) dx = x² sin(x) + 2x cos(x) – 2 ∫ cos(x) dx

∫ x² cos(x) dx = x² sin(x) + 2x cos(x) – 2 sin(x) + C