NIMCET 2017 Question Paper with solutions

Let I = ∫x3 sin x dx
= x3 [- cos x] - ∫ (- cos x) 3x2 dx
−x3cosx + 3 ∫ x2 cos x dx
= −x3 cos x + 3 [x2 (sin x) - ∫ sinx . (2x) dx]
= −x3 cos x + 3x2 sin x - 6 ∫ c sin x dx
= −x3 cos x + 3x2 sin x - 6
[x (- cos x) - ∫ (- cos x) 1 dx]
= x3 cos x + 3x2 sin x + 6x cos x - 6 sin x
∴

π

∫

0

x3 sin x dx = [−x3cosx+3x2sinx+6xcosx−6sinx]0π
= (−π3 cos π + 3π2 sin π + 6π cos π - 6 sin π)
= 90 + 0 + 0 - 6 sin 0)
= π3 + 0 - 6π + 0 + 0 = π3 - 6π