Integrate the function x^2 x^6 + a^6 - Vidyadip

Question

Integrate the function $\frac{{{x^2}}}{{\sqrt {{x^6} + {a^6}} }}$

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Answer

Let $I = \int {\frac{{{x^2}}}{{\sqrt {{x^6} + {a^6}} }}dx} $
$= \frac{1}{3}\int {\frac{{{3x^2}}}{{\sqrt {{{\left( {{x^3}} \right)}^2} + {a^6}} }}dx} $ ...(i)
Putting $x^3 = t$
$\Rightarrow 3{x^2} = \frac{{dt}}{{dx}}$
$ \Rightarrow 3{x^2}dx = dt$
$\therefore$ From eq. (i),
$I = \frac{1}{3}\int \frac{dt}{\sqrt {t^2+a^6}}$
$I = \frac{1}{3}\int \frac{dt}{\sqrt {t^2+(a^3)^2}}$
$= \frac{1}{3}\log \left| {t + \sqrt {{t^2} + {{\left( {{a^3}} \right)}^2}} } \right| + c$
$= \frac{1}{3}\log \left| {{x^3} + \sqrt {{x^6} + {a^6}} } \right| + c$

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