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Q 8  Integrate the functions \frac{x ^ 2 }{\sqrt { x^6 + a ^ 6 }}

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The integration can be down as follows

\frac{x ^ 2 }{\sqrt { x^6 + a ^ 6 }}
let x^3 = t \Rightarrow 3x^2dx =dt

\therefore \frac{x^2}{\sqrt{x^6+a^6}}=\frac{1}{3}\int \frac{dt}{\sqrt{t^2+(a^3)^2}}
                           \\=\frac{1}{3}\log\left | t+\sqrt{t^2+a^6} \right |+C\\ =\frac{1}{3}\log\left | x^3+\sqrt{x^6+a^6} \right |+C........................using \int \frac{dx}{\sqrt{x^2+a^2}} = \log\left | x+\sqrt{x^2+a^2} \right |

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manish

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