integral of ((4-x^2)^{3/2})/(x^6)

解答

\int \frac{( 4 - x ^{2} ) ^{\frac{3}{2}}}{x ^{6}} d x

- \frac{( 4 - x ^{2} ) ^{\frac{3}{2}}}{5 x ^{5}} - \frac{3}{5} - \frac{( - x ^{2} + 4 ) ^{\frac{1}{2}}}{3 x ^{3}} - \frac{1}{3} \cdot \frac{1}{( - + 4 ) ^{\frac{1}{2}}} (- \frac{1}{x}) + C

求解步骤

\int \frac{( 4 - x ^{2} ) ^{\frac{3}{2}}}{x ^{6}} d x

使用分布积分法

= - \frac{( 4 - x ^{2} ) ^{\frac{3}{2}}}{5 x ^{5}} - \int \frac{3 ( - x ^{2} + 4 ) ^{\frac{1}{2}}}{5 x ^{4}} d x

\int \frac{3 ( - x ^{2} + 4 ) ^{\frac{1}{2}}}{5 x ^{4}} d x = \frac{3}{5} - \frac{( - x ^{2} + 4 ) ^{\frac{1}{2}}}{3 x ^{3}} - \frac{1}{3} \cdot \frac{1}{( - + 4 ) ^{\frac{1}{2}}} (- \frac{1}{x})

= - \frac{( 4 - x ^{2} ) ^{\frac{3}{2}}}{5 x ^{5}} - \frac{3}{5} - \frac{( - x ^{2} + 4 ) ^{\frac{1}{2}}}{3 x ^{3}} - \frac{1}{3} \cdot \frac{1}{( - + 4 ) ^{\frac{1}{2}}} (- \frac{1}{x})

解答补常数

= - \frac{( 4 - x ^{2} ) ^{\frac{3}{2}}}{5 x ^{5}} - \frac{3}{5} - \frac{( - x ^{2} + 4 ) ^{\frac{1}{2}}}{3 x ^{3}} - \frac{1}{3} \cdot \frac{1}{( - + 4 ) ^{\frac{1}{2}}} (- \frac{1}{x}) + C

\int \frac{2}{x} y d x

d er i v a t i v e y = \frac{x ^{2} - 2 x + 5}{x ^{2} + 2 x - 3}

\int_{0}^{2} \frac{1}{3 x} d x

t an g e n t f (x) = 9 - 2 x^{2}, (2, 1)

x \to - 5 lim (\frac{x ^{2} + 3 x - 10}{2 x ^{2} + 15 x + 25})