Question

The value of the integral $\int \frac{dx}{x^n(1+x^n{)}^{1/n}},n\in N$ is

• A. $\frac{1}{(1-n)}{(1+\frac{1}{x^n})}^{1-\frac{1}{n}}+C$
• B. $\frac{1}{(1+n)}{(1-\frac{1}{x^n})}^{1+\frac{1}{n}}+C$
• C. $-\frac{1}{(1+n)}{(1-\frac{1}{x^n})}^{1-\frac{1}{n}}+C$
• D. $-\frac{1}{(1+n)}{(1+\frac{1}{x^n})}^{1-\frac{1}{n}}+C$

Answer 1

The correct option is A $\frac{1}{(1-n)}{(1+\frac{1}{x^n})}^{1-\frac{1}{n}}+C$

Consider, $I=\int \frac{dx}{x^n(1+x^n{)}^{1/n}}$

$\Rightarrow I=\int \frac{dx}{x^{n+1}(1+\frac{1}{x^n}{)}^{1/n}}$

Let, $\frac{1}{x^n}=t$ $\Rightarrow \frac{-n}{x^{n+1}}dx=dt$

$\Rightarrow \int \frac{dt}{-n(1+t{)}^{1/n}}=\frac{1}{1-n}{(1+t)}^{1-1/n}=\frac{1}{1-n}{(1+\frac{1}{x^n})}^{1-1/n}$