Question

$\int \frac{1}{x\left(x^n+1\right)}dx$ is equal to :

Answer 1

$\int \frac{dx}{x(x^n+1)}$

Multiply and divide by $x^{n-1}$

$\Rightarrow \int \frac{x^{n-1}dx}{x^{n-1}x(x^n+1)}$

$=\int \frac{x^{n-1}}{x^n(x^n+1)}dx$

Now let $x^n=t$

$n{x}^{n-1}dx=dt$

$\Rightarrow x^{n-1}dx=\frac{dt}{n}$

$\int \frac{d^t}{t(t+1)n}$

$\Rightarrow \frac{1}{n}\int \frac{dt}{t(t+1)}$

Now by partial fraction

$=\frac{1}{n}\int \frac{1}{t}-\frac{1}{t+1}dt$

$=\frac{1}{n}logt-\log (t-1)$

$=\frac{1}{n}log\left(\frac{t}{t+1}\right)=\frac{1}{n}log\frac{x^n}{x^n+1}$$+c$