Question

\[\int\frac{x^2}{x^6 + a^6} dx\]

Answer 1

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