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
 
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
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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[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)