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

Category: QuestionsFind the integral of $$\frac{5x^4}{\sqrt{x^5+9}}$$.
Editor">Editor Staff asked 11 months ago

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
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Editor">Editor Staff answered 11 months ago

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}.