Put x=asin⁡θ. Then dx=acos⁡θdθ∫x2a2−x2dx=∫a2sin2⁡θa2−a2sin2⁡θ⋅acos⁡θdθ=∫a2sin2⁡θa1−sin2⁡θ⋅acos⁡θdθ=∫a2sin2⁡θacos⁡θacos⁡θdθ=a2∫1−cos⁡2θ2dθ=a22∫1dθ−a22∫cos⁡2θdθ=a22(θ)−a22sin⁡2θ2+c=a22Sin−1⁡xa−a242sin⁡θcos⁡θ+c=a22Sin−1⁡xa−a22xa1−x2a2+c=a22Sin−1⁡xa−x2a2−x2+c

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$\int \frac{x^{2}}{\sqrt{a^{2} - x^{2}}} d x =$

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$\frac{a^{2}}{2} {Sin}^{- 1} \frac{x}{a} - \frac{x}{2} \sqrt{a^{2} - x^{2}} + c$

$\frac{a^{2}}{2} {Sin}^{- 1} \frac{x}{a} - \frac{x}{4} \sqrt{a^{2} - x^{2}} + c$

$\frac{a^{2}}{2} {Sin}^{- 1} \frac{x}{a} - \frac{x}{2} \sqrt{a^{2} + x^{2}} + c$

$\frac{a^{2}}{2} {Sinh}^{- 1} \frac{x}{a} + \frac{x}{2} \sqrt{a^{2} - x^{2}} + c$

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Detailed Solution

Put $x = a \sin θ . Then d x = a \cos θ d θ$

$\int \frac{x^{2}}{\sqrt{a^{2} - x^{2}}} d x = \int \frac{a^{2} \sin^{2} θ}{\sqrt{a^{2} - a^{2} \sin^{2} θ}} \cdot a \cos θ d θ = \int \frac{a^{2} \sin^{2} θ}{a \sqrt{1 - \sin^{2} θ}} \cdot a \cos θ d θ$

$= \int \frac{a^{2} \sin^{2} θ}{a \cos θ} a \cos θ d θ = a^{2} \int (\frac{1 - \cos 2 θ}{2}) d θ = \frac{a^{2}}{2} \int 1 d θ - \frac{a^{2}}{2} \int \cos 2 θ d θ$

$\begin{array}{l} = \frac{a^{2}}{2} (θ) - \frac{a^{2}}{2} \frac{\sin 2 θ}{2} + c = \frac{a^{2}}{2} {Sin}^{- 1} (\frac{x}{a}) - \frac{a^{2}}{4} 2 \sin θ \cos θ + c \\ = \frac{a^{2}}{2} {Sin}^{- 1} (\frac{x}{a}) - \frac{a^{2}}{2} (\frac{x}{a}) \sqrt{1 - \frac{x^{2}}{a^{2}}} + c = \frac{a^{2}}{2} {Sin}^{- 1} (\frac{x}{a}) - \frac{x}{2} \sqrt{a^{2} - x^{2}} + c \end{array}$