Question

Find ∫ 2x^3 e^x^2 dx.

(a) -e^x^2 (x^2+2)+C

(b) e^x^2 (x^2-1)+C

(c) 2e^x^2 (x^2+1)+C

(d) e^x^2 (x-1)+C

This question was addressed to me in an interview for job.

This interesting question is from Integration by Parts topic in division Integrals of Mathematics – Class 12

Answer 1

Right choice is (b) e^x^2 (x^2-1)+C

The explanation: Let x^2=t

Differentiating w.r.t x, we get

2x dx=dt

∴∫ 2x^2 e^x^2 dx=∫ te^t dt

By using the formula, ∫ u.v dx=u∫ v dx-∫ u’ (∫ v dx) ,we get

∫ t e^t dt=t∫ e^t dt-∫ (t)’∫ e^t dt

=te^t-∫ e^t dt

=te^t-e^t=e^t (t-1)

Replacing t with x^2, we get

∫ 2x^3 e^x^2 dx=e^x^2 (x^2-1)+C