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

解答

\int \frac{x ^{- 2} + x ^{- 3}}{x} d x

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

求解步骤

\int \frac{x ^{- 2} + x ^{- 3}}{x} d x

化简

\frac{x ^{- 2} + x ^{- 3}}{x} : \frac{x + 1}{x ^{4}}

= \int \frac{x + 1}{x ^{4}} d x

乘开

\frac{x + 1}{x ^{4}} : \frac{1}{x ^{3}} + \frac{1}{x ^{4}}

= \int \frac{1}{x ^{3}} + \frac{1}{x ^{4}} d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{x ^{3}} d x + \int \frac{1}{x ^{4}} d x

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

\int \frac{1}{x ^{4}} d x = - \frac{1}{3 x ^{3}}

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

解答补常数

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