integral of x^3(1-x^2)^{-1/2}

解答

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

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

求解步骤

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

化简

x^{3} (1 - x^{2})^{- \frac{1}{2}} : \frac{x ^{3}}{( 1 - x ^{2} ) ^{\frac{1}{2}}}

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

使用分布积分法

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

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

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

化简

= \frac{1}{3} (1 - x^{2})^{\frac{3}{2}} - 1 - x^{2}

解答补常数

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

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