Practicing Success
The value of $\int \frac{1}{x^n\left(1+x^n\right)^{1 / n}} d x,(n \in N)$, is |
$\frac{1}{1-n}\left\{1+\frac{1}{x^n}\right\}^{1-\frac{1}{n}}+C$ $\frac{1}{1+n}\left\{1-\frac{1}{x^n}\right\}^{1-\frac{1}{n}}+C$ $-\frac{1}{1-n}\left\{1-\frac{1}{x^n}\right\}^{1-\frac{1}{n}}+C$ $-\frac{1}{1+n}\left\{1+\frac{1}{x^n}\right\}^{1-\frac{1}{n}}+C$ |
$\frac{1}{1-n}\left\{1+\frac{1}{x^n}\right\}^{1-\frac{1}{n}}+C$ |
Let $I =\int \frac{1}{x^n\left(1+x^n\right)^{1 / n}} d x=\int \frac{1}{x^{n+1}\left(1+\frac{1}{x^n}\right)^{1 / n}} d x$ $\Rightarrow I =\int\left(1+x^{-n}\right)^{-1 / n} x^{-n-1} d x$ $\Rightarrow I =-\frac{1}{n} \int\left(1+x^{-n}\right)^{-1 / n} d\left(1+x^{-n}\right)$ $\Rightarrow I=-\frac{1}{n} \frac{\left(1+x^{-n}\right)^{-\frac{1}{n}+1}}{\left(-\frac{1}{n}+1\right)}+C=\frac{1}{1-n}\left\{1+\frac{1}{x^n}\right\}^{1-\frac{1}{n}}+C$ |