Question

` ∫  { x^2 dx}/{x^6 - a^6} dx `

Answer 1

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