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受欢迎的微积分 >

integral of (x^3)/(\sqrt[9]{x^2+5)}

解答

\int \frac{x ^{3}}{9 x ^{2} + 5} d x

解答

\frac{1}{2} (\frac{9}{17} (x^{2} + 5)^{\frac{17}{9}} - \frac{45}{8} (x^{2} + 5)^{\frac{8}{9}}) + C

求解步骤

\int \frac{x ^{3}}{9 x ^{2} + 5} d x

使用换元积分法

= \int \frac{u - 5}{2 9 u} d u

提出常数:

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

= \frac{1}{2} \cdot \int \frac{u - 5}{9 u} d u

乘开

\frac{u - 5}{9 u} : u^{\frac{8}{9}} - \frac{5}{9 u}

= \frac{1}{2} \cdot \int u^{\frac{8}{9}} - \frac{5}{9 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

= \frac{1}{2} (\int u^{\frac{8}{9}} d u - \int \frac{5}{9 u} d u)

\int u^{\frac{8}{9}} d u = \frac{9}{17} u^{\frac{17}{9}}

\int \frac{5}{9 u} d u = \frac{45}{8} u^{\frac{8}{9}}

= \frac{1}{2} (\frac{9}{17} u^{\frac{17}{9}} - \frac{45}{8} u^{\frac{8}{9}})

u = x^{2} + 5

代回

= \frac{1}{2} (\frac{9}{17} (x^{2} + 5)^{\frac{17}{9}} - \frac{45}{8} (x^{2} + 5)^{\frac{8}{9}})

解答补常数

= \frac{1}{2} (\frac{9}{17} (x^{2} + 5)^{\frac{17}{9}} - \frac{45}{8} (x^{2} + 5)^{\frac{8}{9}}) + C

作图

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