Question

$I=\int \frac{1}{x(x^6+1)}dx$

Answer 1

$I=\int \frac{1}{x\left(x^6+1\right)}dx$

putting $x^6+1=t$

$6x^{5}dx=dt$

$dx=\frac{dt}{6x^5}$

$I=\int \frac{dt}{6x^5\times x\times t}$

$I=\frac{1}{6}\int \frac{dt}{t\left(t-1\right)}$

$I=\frac{1}{6}\int \left(\frac{1}{t-1}-\frac{1}{t}\right)dt$

$I=\frac{1}{6}\int \frac{1}{t-1}dt-\frac{1}{6}\int \frac{1}{t}dt$

$I=\frac{1}{6}\log (t-1)-\frac{1}{6}\log \left(t\right)+C$

$I=\frac{1}{6}\log \left(\frac{t-1}{t}\right)+C$

$I=\frac{1}{6}\log \left(\frac{x^6}{x^6+1}\right)+C$