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Need Solution for R.D.Sharma Maths Class 12 Chapter 18 Indefinite Integrals Exercise Revision Exercise Question 101 Maths Textbook Solution.

Answers (1)

Answer: $\frac{1}{n} \log \left|\frac{\sqrt{1+x^{n}}-1}{\sqrt{1+x^{n}}+1}\right|+C$

Hint

Given: $\int \frac{1}{x \sqrt{1+x^{n}}} d x$

Solution:

$I=\int \frac{d x}{x \sqrt{1+x^{n}}}$

$=\int \frac{x^{n-1} d x}{x^{n-1} x^{1} \sqrt{1+x^{n}}}$

$=\int \frac{x^{n-1} d x}{x^{n} \sqrt{1+x^{n}}}$

$\text { putting } x^{n}=t$

$\Rightarrow n x^{n-1} d x=d t$

$\Rightarrow x^{n-1} d x=\frac{d t}{n}$

$I=\frac{1}{n} \int \frac{d t}{t \sqrt{1+t}}$

$\operatorname{let} 1+t=p^{2}$

$d t=2 p d p$

$I=\frac{1}{n} \int \frac{2 p d p}{\left(p^{2}-1\right) p}$

$=\frac{2}{n} \int \frac{d p}{p^{2}-1^{2}}$

$=\frac{2}{n} \times \frac{1}{2} \log \left|\frac{p-1}{p+1}\right|+C$

$=\frac{1}{n} \log \left|\frac{\sqrt{1+t}-1}{\sqrt{1+t}+1}\right|+C$

$=\frac{1}{n} \log \left|\frac{\sqrt{1+x^{n}}-1}{\sqrt{1+x^{n}}+1}\right|+C$

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