integral of (9-5x)^7

解答

\int (9 - 5 x)^{7} d x

- \frac{1}{40} (9 - 5 x)^{8} + C

求解步骤

\int (9 - 5 x)^{7} d x

使用换元积分法

= \int - \frac{u ^{7}}{5} d u

提出常数:

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

= - \frac{1}{5} \cdot \int u^{7} d u

使用幂法则

= - \frac{1}{5} \cdot \frac{u ^{8}}{8}

u = 9 - 5 x

代回

= - \frac{1}{5} \cdot \frac{( 9 - 5 x ) ^{8}}{8}

化简

- \frac{1}{5} \cdot \frac{( 9 - 5 x ) ^{8}}{8} : - \frac{1}{40} (9 - 5 x)^{8}

= - \frac{1}{40} (9 - 5 x)^{8}

解答补常数

= - \frac{1}{40} (9 - 5 x)^{8} + C