Question

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

Answer 1

$\int \left(\frac{1}{x(x^n+1)}\right)dx=\int \left(\frac{x^{n-1}}{x^n(x^n+1)}\right)dxlet{x}^n=tn{x}^{n-1}dx=dt\therefore x^{n-1}dx=\left(\frac{dt}{n}\right)\therefore I=\left(\frac{1}{n}\right)\int \left(\frac{1}{t(t+1)}dt\right)=\left(\frac{1}{n}\right)[\int (\frac{1}{t})dt-\int (\frac{1}{t+1}\left)dt\right]=\left(\frac{1}{n}\right)[logt-log(t+1\left)\right]+c=\left(\frac{1}{n}\right)log\left(\right(\frac{1}{n}\left)\right)+c=\left(\frac{1}{n}\right)log\left(\right(\frac{x^n}{x^n+1}\left)\right)+c$