Question

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

Answer 1

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

\[ = \int\frac{x^5 dx}{x^6 \left( x^6 + 1 \right)}\]
\[\text{ let }x^6 = t\]
\[ \Rightarrow 6 x^5 dx = dt\]

\[ \Rightarrow x^5 dx = \frac{dt}{6}\]
\[Now, \int\frac{dx}{x^6 \left( x^6 + 1 \right)}\]
\[ = \frac{1}{6}\int\frac{dt}{t\left( t + 1 \right)}\]
\[ = \frac{1}{6}\int\frac{dt}{t^2 + t}\]
\[ = \frac{1}{6}\int\frac{dt}{t^2 + t + \frac{1}{4} - \frac{1}{4}}\]
\[ = \frac{1}{6}\int\frac{dt}{\left( t + \frac{1}{2} \right)^2 - \left( \frac{1}{2} \right)^2}\]
\[ = \frac{1}{6} \times \frac{1}{2 \times \frac{1}{2}} \text{ log  }\left| \frac{t + \frac{1}{2} - \frac{1}{2}}{t + \frac{1}{2} + \frac{1}{2}} \right| + C\]
\[ = \frac{1}{6} \text{ log }  \left| \frac{t}{t + 1} \right| + C\]
\[ = \frac{1}{6} \text{ log  }\left| \frac{x^6}{x^6 + 1} \right| + C\]