integral of ((x^{-1/3}-5))/x

解答

\int \frac{( x ^{- \frac{1}{3}} - 5 )}{x} d x

- \frac{3}{x ^{\frac{1}{3}}} - 5 ln ∣ x ∣ + C

求解步骤

\int \frac{x ^{- \frac{1}{3}} - 5}{x} d x

乘开

\frac{x ^{- \frac{1}{3}} - 5}{x} : \frac{1}{x ^{\frac{4}{3}}} - \frac{5}{x}

= \int \frac{1}{x ^{\frac{4}{3}}} - \frac{5}{x} 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 ^{\frac{4}{3}}} d x - \int \frac{5}{x} d x

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

\int \frac{5}{x} d x = 5 ln ∣ x ∣

= - \frac{3}{x ^{\frac{1}{3}}} - 5 ln ∣ x ∣

解答补常数

= - \frac{3}{x ^{\frac{1}{3}}} - 5 ln ∣ x ∣ + C

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