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受欢迎的微积分 >

integral of 1/(x^3\sqrt[5]{1+1/x)}

解答

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

解答

- \frac{5}{9} (\frac{x + 1}{x})^{\frac{9}{5}} + \frac{5}{4} (\frac{x + 1}{x})^{\frac{4}{5}} + C

求解步骤

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

使用换元积分法

= \int - \frac{u - 1}{5 u} d u

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int \frac{u - 1}{5 u} d u

乘开

\frac{u - 1}{5 u} : u^{\frac{4}{5}} - \frac{1}{5 u}

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

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 u^{\frac{4}{5}} d u - \int \frac{1}{5 u} d u)

\int u^{\frac{4}{5}} d u = \frac{5}{9} u^{\frac{9}{5}}

\int \frac{1}{5 u} d u = \frac{5}{4} u^{\frac{4}{5}}

= - (\frac{5}{9} u^{\frac{9}{5}} - \frac{5}{4} u^{\frac{4}{5}})

u = 1 + \frac{1}{x}

代回

= - (\frac{5}{9} (1 + \frac{1}{x})^{\frac{9}{5}} - \frac{5}{4} (1 + \frac{1}{x})^{\frac{4}{5}})

化简

- (\frac{5}{9} (1 + \frac{1}{x})^{\frac{9}{5}} - \frac{5}{4} (1 + \frac{1}{x})^{\frac{4}{5}}) : - \frac{5}{9} (\frac{x + 1}{x})^{\frac{9}{5}} + \frac{5}{4} (\frac{x + 1}{x})^{\frac{4}{5}}

= - \frac{5}{9} (\frac{x + 1}{x})^{\frac{9}{5}} + \frac{5}{4} (\frac{x + 1}{x})^{\frac{4}{5}}

解答补常数

= - \frac{5}{9} (\frac{x + 1}{x})^{\frac{9}{5}} + \frac{5}{4} (\frac{x + 1}{x})^{\frac{4}{5}} + C

作图

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