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受欢迎的微积分 >

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

解答

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

解答

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

求解步骤

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

分解

x^{3} + x^{2} : x^{2} x + 1

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

x^{2} = (x),

假定

x \geq 0

= \int x x + 1 d x

使用换元积分法

= \int (u - 1) u d u

乘开

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

= \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

= \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{2}{5} u^{\frac{5}{2}} - \frac{2}{3} u^{\frac{3}{2}}

u = x + 1

代回

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

解答补常数

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

作图

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