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受欢迎的微积分 >

integral of x^3*sqrt(x^2+1)

解答

\int x^{3} \cdot x^{2} + 1 d x

解答

\frac{1}{15} (3 x^{2} - 2) (x^{2} + 1)^{\frac{3}{2}} + C

求解步骤

\int x^{3} x^{2} + 1 d x

使用换元积分法

= \int \frac{( u - 1 ) u}{2} 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 (u - 1) u d u

乘开

(u - 1) u : u^{\frac{3}{2}} - u

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

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

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

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

u = x^{2} + 1

代回

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

化简

\frac{1}{2} (\frac{2}{5} (x^{2} + 1)^{\frac{5}{2}} - \frac{2}{3} (x^{2} + 1)^{\frac{3}{2}}) : \frac{1}{15} (3 x^{2} - 2) (x^{2} + 1)^{\frac{3}{2}}

= \frac{1}{15} (3 x^{2} - 2) (x^{2} + 1)^{\frac{3}{2}}

解答补常数

= \frac{1}{15} (3 x^{2} - 2) (x^{2} + 1)^{\frac{3}{2}} + C

作图

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