Question

Find \(\int_2^3 \,2x^2 \,e^{x^3} \,dx\).

(a) \(e^{27}-e^8\)

(b) \(\frac{2}{3} (e^{27}-e^8)\)

(c) \(\frac{2}{3} (e^8-e^{27})\)

(d) \(\frac{2}{3} (e^{27}+e^8)\)

This question was addressed to me during a job interview.

I would like to ask this question from Evaluation of Definite Integrals by Substitution topic in section Integrals of Mathematics – Class 12

Answer 1

Correct choice is (b) \(\frac{2}{3} (e^{27}-e^8)\)

Explanation: I=\(\int_2^3 \,2x^2 \,e^{x^3} \,dx\)

Let x^3=t

Differentiating w.r.t x, we get

3x^2 dx=dt

x^2 dx=\(\frac{dt}{3}\)

The new limits

When x=2, t=8

When x=3, t=27

∴\(\int_2^3 \,2x^2 \,e^{x^3} \,dx=\frac{2}{3} \int_8^{27} \,e^t \,dt\)

=\(\frac{2}{3} [e^t]_8^{27}=\frac{2}{3} (e^{27}-e^8).\)