Question

Evaluate \(\int_{\sqrt{2}}^2 \,14x \,log⁡ x^2 \,dx\)

(a) 14(3 log⁡2-1)

(b) 14(3 log⁡2+1)

(c) log⁡2-1

(d) 3 log⁡2-1

The question was posed to me in an interview for internship.

My enquiry is from Evaluation of Definite Integrals by Substitution topic in division Integrals of Mathematics – Class 12

Answer 1

Right answer is (a) 14(3 log⁡2-1)

Explanation: I=\(\int_{\sqrt{2}}^2 \,14x \,log⁡ x^2 \,dx\)

Let x^2=t

Differentiating w.r.t x, we get

2x dx=dt

The new limits

When x=\(\sqrt{2}\), t=2

When x=2, t=4

∴\(\int_{\sqrt{2}}^2 \,14x \,log⁡ x^2 \,dx =\int_2^4 \,7 \,log⁡ t \,dt\)

Using integration by parts, we get

\(\int_2^4 \,7 \,log⁡ t \,dt=7(log⁡ t\int dt-\int (log⁡t)’ \int \,dt)\)

=7 (t log⁡t-t)2^4

=7(4 log⁡4-4-2 log⁡2+2)

=7(6 log⁡2-2)=14(3 log⁡2-1)