integral of (x^8)/(sqrt(1+x^3))

解答

\int \frac{x ^{8}}{1 + x ^{3}} d x

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

求解步骤

\int \frac{x ^{8}}{1 + x ^{3}} d x

使用换元积分法

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

乘开

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

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

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

\int 2 u d u = \frac{4}{3} u^{\frac{3}{2}}

\int \frac{1}{u} d u = 2 u

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

u = 1 + x^{3}

代回

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

解答补常数

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