Question

$\int \frac{dx}{x^{n+1}{(1+x^n)}^{1/n}}\forall n\epsilon$ N is

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

Answer 1

The correct option is A $\frac{1}{1-n}{[1+\frac{1}{x^n}]}^{1-\frac{1}{n}}+c$

Let $t={(1+x^{-n})}^{1-\frac{1}{n}}$
$\Rightarrow dt=-(n-1){(1+x^{-n})}^{-\frac{1}{n}}\cdot x^{-n-1}dx$
Now,
$\int \frac{dx}{x^{n+1}{(1+x^n)}^{1/n}}=\int \frac{dt}{n-1}=\frac{t}{n-1}+c$
$=\frac{{(1+x^{-n})}^{1-\frac{1}{n}}}{n-1}+c$