Question

The value of $\int \frac{dx}{x^n(1+x^n{)}^{\frac{1}{n}}},nϵN$

• 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 D $\frac{1}{1-n}{\{1+\frac{1}{x^n}\}}^{1-\frac{1}{n}}+c$

$I=\int \frac{dx}{x^n(1+x^n{)}^{1/n}}=\int \frac{dx}{x^{n+1}{(1+\frac{1}{x^n})}^{1/n}}Let1+\frac{1}{x^n}=t\Rightarrow \frac{-n}{x^{n+1}}dx=dt\therefore I=-\frac{1}{n}\int \frac{dt}{t^{\frac{1}{n}}}=-\frac{1}{n\left[\frac{t^{-\frac{1}{n}+1}}{-\frac{1}{n}+1}\right]}+c=-\frac{{(1+\frac{1}{x^n})}^{1-\frac{1}{n}}}{n-1}+c$