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受欢迎的微积分 >

integral of x^7sqrt(x^4+3)

解答

\int x^{7} x^{4} + 3 d x

解答

\frac{1}{10} (x^{4} - 2) (x^{4} + 3)^{\frac{3}{2}} + C

求解步骤

\int x^{7} x^{4} + 3 d x

使用换元积分法

= \int \frac{( u - 3 ) u}{4} d u

提出常数:

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

= \frac{1}{4} \cdot \int (u - 3) u d u

乘开

(u - 3) u : u^{\frac{3}{2}} - 3 u

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

\int u^{\frac{3}{2}} d u = \frac{2}{5} u^{\frac{5}{2}}

\int 3 u d u = 2 u^{\frac{3}{2}}

= \frac{1}{4} (\frac{2}{5} u^{\frac{5}{2}} - 2 u^{\frac{3}{2}})

u = x^{4} + 3

代回

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

化简

\frac{1}{4} (\frac{2}{5} (x^{4} + 3)^{\frac{5}{2}} - 2 (x^{4} + 3)^{\frac{3}{2}}) : \frac{1}{10} (x^{4} - 2) (x^{4} + 3)^{\frac{3}{2}}

= \frac{1}{10} (x^{4} - 2) (x^{4} + 3)^{\frac{3}{2}}

解答补常数

= \frac{1}{10} (x^{4} - 2) (x^{4} + 3)^{\frac{3}{2}} + C

作图

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