Question

Find the integral of \(\frac{5x^4}{\sqrt{x^5+9}}\).

(a) \(\sqrt{x^5+9}\)

(b) \(2\sqrt{x^5-9}\)

(c) 2(x^5+9)

(d) \(2\sqrt{x^5+9}\)

I have been asked this question in examination.

This intriguing question comes from Methods of Integration-1 in portion Integrals of Mathematics – Class 12

Answer 1

Correct option is (d) \(2\sqrt{x^5+9}\)

For explanation I would say: Let x^5+9=t

Differentiating w.r.t x, we get

5x^4 dx=dt

\(\int \frac{5x^4}{\sqrt{x^5+9}} dx=\int \frac{dt}{\sqrt{t}}\)

=\(\frac{t^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}=2\sqrt{t}\)

Replacing t with x^5+9, we get

\(\int \frac{5x^4}{\sqrt{x^5+9}} dx=2\sqrt{x^5+9}\).