Question

Find \(\int_{-1}^1 \frac{5x^4}{\sqrt{x^5+3}} dx\).

(a) 4-\(\sqrt{2}\)

(b) 4+2\(\sqrt{2}\)

(c) 4-2\(\sqrt{2}\)

(d) 1-2\(\sqrt{2}\)

This question was posed to me in an online quiz.

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

Answer 1

Correct choice is (c) 4-2\(\sqrt{2}\)

To explain I would say: I=\(\int_{-1}^1 \frac{5x^4}{\sqrt{x^5+3}} dx\)

Let x^5+3=t

Differentiating w.r.t x, we get

5x^4 dx=dt

The new limits

when x=-1,t=2

when x=1,t=4

∴\(\int_{-1}^1 \frac{5x^4}{\sqrt{x^5+3}} dx=\int_2^4 \frac{dt}{\sqrt{t}}\)

=\([2\sqrt{t}]_2^4=2(\sqrt{4}-\sqrt{2})=4-2\sqrt{2}\)