Question

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

Answer 1

$$\begin{gathered} \int \frac{dx}{x\left(x^6+1\right)} \\ =\int \frac{x^{5}dx}{x^6\left(x^6+1\right)} \\ \mathrm{let}{x}^6=t \\ \Rightarrow 6x^{5}dx=dt \\ \Rightarrow x^{5}dx=\frac{dt}{6} \\ \mathrm{Now},\int \frac{dx}{x^6\left(x^6+1\right)} \\ =\frac{1}{6}\int \frac{dt}{t\left(t+1\right)} \\ =\frac{1}{6}\int \frac{dt}{t^2+t} \\ =\frac{1}{6}\int \frac{dt}{t^2+t+{\displaystyle \frac{1}{4}}-{\displaystyle \frac{1}{4}}} \\ =\frac{1}{6}\int \frac{dt}{{\left(t+{\displaystyle \frac{1}{2}}\right)}^2-{\left({\displaystyle \frac{1}{2}}\right)}^2} \\ =\frac{1}{6}\times \frac{1}{2\times {\displaystyle \frac{1}{2}}}\log \left|\frac{t+{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2}}}{t+{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}}\right|+C \\ =\frac{1}{6}\log \left|\frac{t}{t+1}\right|+C \\ =\frac{1}{6}\log \left|\frac{x^6}{x^6+1}\right|+C \end{gathered}$$