Question

\[\int\frac{3 x^5}{1 + x^{12}} dx\]

Answer 1

\[\int\frac{3 x^5}{1 + x^{12}}dx\]
\[\text{ let } x^6 = t\]
\[ \Rightarrow 6 x^5 dx = dt\]
\[ \Rightarrow x^5 dx = \frac{dt}{6}\]
\[Now, \int\frac{3 x^5}{1 + x^{12}}dx\]
\[ = \frac{3}{6}\int\frac{dt}{1 + t^2}\]
\[ = \frac{1}{2} \tan^{- 1} \left( t \right) + C\]

\[= \frac{1}{2} \tan^{- 1} \left( x^6 \right) + C\]