Question

Find \(\int_1^2 \frac{12 \,log⁡x}{x} \,dx\).

(a) -12 log⁡2

(b) 24 log⁡2

(c) 12 log⁡2

(d) 24 log⁡4

This question was posed to me in an interview.

This key question is from Evaluation of Definite Integrals by Substitution in portion Integrals of Mathematics – Class 12

Answer 1

Right choice is (b) 24 log⁡2

To explain I would say: I=\(\int_1^2 \frac{12 log⁡x}{x} \,dx\)

Let log⁡x=t

Differentiating w.r.t x, we get

\(\frac{1}{x} \,dx=dt\)

The new limits

When x=1,t=0

When x=2,t=log⁡2

\(\int_1^2 \frac{12 log⁡x}{x} dx=12\int_0^{log⁡2} \,t \,dt\)

=\(12[t^2]_0^{log⁡2}=12((log⁡2)^2-0)\)

=12 log⁡4=24 log⁡2(∵(log⁡2)^2=log⁡2.log⁡2=log⁡4=2 log⁡2)