integral of x(9x^2-5)^{3/2}

解答

\int x (9 x^{2} - 5)^{\frac{3}{2}} d x

\frac{1}{45} (9 x^{2} - 5)^{\frac{5}{2}} + C

求解步骤

\int x (9 x^{2} - 5)^{\frac{3}{2}} d x

使用换元积分法

= \int \frac{u ^{\frac{3}{2}}}{18} d u

提出常数:

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

= \frac{1}{18} \cdot \int u^{\frac{3}{2}} d u

使用幂法则

= \frac{1}{18} \cdot \frac{2}{5} u^{\frac{5}{2}}

u = 9 x^{2} - 5

代回

= \frac{1}{18} \cdot \frac{2}{5} (9 x^{2} - 5)^{\frac{5}{2}}

化简

\frac{1}{18} \cdot \frac{2}{5} (9 x^{2} - 5)^{\frac{5}{2}} : \frac{1}{45} (9 x^{2} - 5)^{\frac{5}{2}}

= \frac{1}{45} (9 x^{2} - 5)^{\frac{5}{2}}

解答补常数

= \frac{1}{45} (9 x^{2} - 5)^{\frac{5}{2}} + C