Question
Evaluate the following integrals:
$\int\text{x}^2\sqrt{\text{a}^6-\text{x}^6}\text{dx}$
$\int\text{x}^2\sqrt{\text{a}^6-\text{x}^6}\text{dx}$
✓
Answer
Let $\text{I}=\int\text{x}^2\sqrt{\text{a}^6-\text{x}^6}\text{dx}$
Let $\text{x}^3=\text{t}$
$\Rightarrow3\text{x}^2\text{dx}=\text{dt}$
$\therefore\ \text{I}=\frac{1}{3}\int\sqrt{\text{a}^6-\text{t}^2}\text{dt}$
$=\frac{1}{3}\begin{Bmatrix}\frac{\text{t}}{2}\sqrt{\text{a}^6-\text{t}^2}+\frac{\text{a}^6}{2}\sin^{-1}\Big(\frac{\text{t}}{\text{a}^3}\Big)\end{Bmatrix}+\text{C}$
$\therefore\ \text{I}=\frac{\text{x}^3}{6}\sqrt{\text{a}^6-\text{x}^6}+\frac{\text{a}^6}{6}\sin^{-1}\Big(\frac{\text{x}^3}{\text{a}^3}\Big)+\text{C}$
Let $\text{x}^3=\text{t}$
$\Rightarrow3\text{x}^2\text{dx}=\text{dt}$
$\therefore\ \text{I}=\frac{1}{3}\int\sqrt{\text{a}^6-\text{t}^2}\text{dt}$
$=\frac{1}{3}\begin{Bmatrix}\frac{\text{t}}{2}\sqrt{\text{a}^6-\text{t}^2}+\frac{\text{a}^6}{2}\sin^{-1}\Big(\frac{\text{t}}{\text{a}^3}\Big)\end{Bmatrix}+\text{C}$
$\therefore\ \text{I}=\frac{\text{x}^3}{6}\sqrt{\text{a}^6-\text{x}^6}+\frac{\text{a}^6}{6}\sin^{-1}\Big(\frac{\text{x}^3}{\text{a}^3}\Big)+\text{C}$
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