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
NCERT Solutions for Subject Clas 12 Math Select the correct answer from above options 
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1 Answers
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}.