integral of x^2(a^2-x^2)^{-3/2}

解

\int x^{2} (a^{2} - x^{2})^{- \frac{3}{2}} d x

\frac{x ^{3}}{a ^{2} ( - x ^{2} + a ^{2} ) ^{\frac{1}{2}}} - arcsin (\frac{x}{a}) + \frac{1}{2} sin (2 arcsin (\frac{x}{a})) + C

解答ステップ

\int x^{2} (a^{2} - x^{2})^{- \frac{3}{2}} d x

簡素化

x^{2} (a^{2} - x^{2})^{- \frac{3}{2}} : \frac{x ^{2}}{( a ^{2} - x ^{2} ) ^{\frac{3}{2}}}

= \int \frac{x ^{2}}{( a ^{2} - x ^{2} ) ^{\frac{3}{2}}} d x

部分積分を適用する

= \frac{x ^{3}}{a ^{2} ( - x ^{2} + a ^{2} ) ^{\frac{1}{2}}} - \int \frac{2 x ^{2}}{a ^{2} ( - x ^{2} + a ^{2} ) ^{\frac{1}{2}}} d x

\int \frac{2 x ^{2}}{a ^{2} ( - x ^{2} + a ^{2} ) ^{\frac{1}{2}}} d x = arcsin (\frac{x}{a}) - \frac{1}{2} sin (2 arcsin (\frac{x}{a}))

= \frac{x ^{3}}{a ^{2} ( - x ^{2} + a ^{2} ) ^{\frac{1}{2}}} - (arcsin (\frac{x}{a}) - \frac{1}{2} sin (2 arcsin (\frac{x}{a})))

簡素化

= \frac{x ^{3}}{a ^{2} ( - x ^{2} + a ^{2} ) ^{\frac{1}{2}}} - arcsin (\frac{x}{a}) + \frac{1}{2} sin (2 arcsin (\frac{x}{a}))

定数を解答に追加する

= \frac{x ^{3}}{a ^{2} ( - x ^{2} + a ^{2} ) ^{\frac{1}{2}}} - arcsin (\frac{x}{a}) + \frac{1}{2} sin (2 arcsin (\frac{x}{a})) + C