# Find $$\int_1^2 \frac{12 \,log⁡x}{x} \,dx$$.

Category: QuestionsFind $$\int_1^2 \frac{12 \,log⁡x}{x} \,dx$$.
Editor">Editor Staff asked 11 months ago

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

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This key question is from Evaluation of Definite Integrals by Substitution in portion Integrals of Mathematics – Class 12
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Editor">Editor Staff answered 11 months ago

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)