integral of (sqrt((9-x^2)^3))/(x^6)

解答

\int \frac{( 9 - x ^{2} ) ^{3}}{x ^{6}} d x

- \frac{( 9 - x ^{2} ) ^{\frac{3}{2}}}{5 x ^{5}} + \frac{- x ^{2} + 9}{5 x ^{3}} - \frac{9 - x ^{2}}{45 x} + C

求解步骤

\int \frac{( 9 - x ^{2} ) ^{3}}{x ^{6}} d x

(9 - x^{2})^{3} = (9 - x^{2})^{\frac{3}{2}},

假定

(9 - x^{2}) \geq 0

= \int \frac{( 9 - x ^{2} ) ^{\frac{3}{2}}}{x ^{6}} d x

使用分布积分法

= - \frac{( 9 - x ^{2} ) ^{\frac{3}{2}}}{5 x ^{5}} - \int \frac{3 - x ^{2} + 9}{5 x ^{4}} d x

\int \frac{3 - x ^{2} + 9}{5 x ^{4}} d x = - \frac{- x ^{2} + 9}{5 x ^{3}} + \frac{9 - x ^{2}}{45 x}

= - \frac{( 9 - x ^{2} ) ^{\frac{3}{2}}}{5 x ^{5}} - (- \frac{- x ^{2} + 9}{5 x ^{3}} + \frac{9 - x ^{2}}{45 x})

化简

= - \frac{( 9 - x ^{2} ) ^{\frac{3}{2}}}{5 x ^{5}} + \frac{- x ^{2} + 9}{5 x ^{3}} - \frac{9 - x ^{2}}{45 x}

解答补常数

= - \frac{( 9 - x ^{2} ) ^{\frac{3}{2}}}{5 x ^{5}} + \frac{- x ^{2} + 9}{5 x ^{3}} - \frac{9 - x ^{2}}{45 x} + C

\int 30 + 0.2 x d x

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