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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1 Answers
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 log⁡4=24 log⁡2(∵(log⁡2)^2=log⁡2.log⁡2=log⁡4=2 log⁡2)