integral of x^5*sqrt(x^3+1)

解答

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

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

求解步骤

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

使用换元积分法

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

提出常数:

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

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

乘开

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

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

u = x^{3} + 1

代回

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

解答补常数

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

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