integral of (x^2)/((7-x^3)^2)

解答

\int \frac{x ^{2}}{( 7 - x ^{3} ) ^{2}} d x

\frac{1}{3 ( 7 - x ^{3} )} + C

求解步骤

\int \frac{x ^{2}}{( 7 - x ^{3} ) ^{2}} d x

使用换元积分法

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

使用幂法则

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

u = 7 - x^{3}

代回

= - \frac{1}{3} (- \frac{1}{7 - x ^{3}})

化简

- \frac{1}{3} (- \frac{1}{7 - x ^{3}}) : \frac{1}{3 ( 7 - x ^{3} )}

= \frac{1}{3 ( 7 - x ^{3} )}

解答补常数

= \frac{1}{3 ( 7 - x ^{3} )} + C

\int (u + u^{4}) d u

\int_{1}^{1000} \frac{33}{x ^{3}} d x

\int_{- 5}^{5} (25 - x^{2}) d x

\int \frac{x}{16} d x

\frac{\partial}{\partial x} (\frac{2 x + y}{y - x})