integral of s(5s^2-8)^6

解答

\int s (5 s^{2} - 8)^{6} d s

\frac{1}{70} (5 s^{2} - 8)^{7} + C

求解步骤

\int s (5 s^{2} - 8)^{6} d s

使用换元积分法

= \int \frac{u ^{6}}{10} d u

提出常数:

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

= \frac{1}{10} \cdot \int u^{6} d u

使用幂法则

= \frac{1}{10} \cdot \frac{u ^{7}}{7}

u = 5 s^{2} - 8

代回

= \frac{1}{10} \cdot \frac{( 5 s ^{2} - 8 ) ^{7}}{7}

化简

\frac{1}{10} \cdot \frac{( 5 s ^{2} - 8 ) ^{7}}{7} : \frac{1}{70} (5 s^{2} - 8)^{7}

= \frac{1}{70} (5 s^{2} - 8)^{7}

解答补常数

= \frac{1}{70} (5 s^{2} - 8)^{7} + C