integral of (5-3x)^8

解答

\int (5 - 3 x)^{8} d x

- \frac{1}{27} (5 - 3 x)^{9} + C

求解步骤

\int (5 - 3 x)^{8} d x

使用换元积分法

= \int - \frac{u ^{8}}{3} 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 u^{8} d u

使用幂法则

= - \frac{1}{3} \cdot \frac{u ^{9}}{9}

u = 5 - 3 x

代回

= - \frac{1}{3} \cdot \frac{( 5 - 3 x ) ^{9}}{9}

化简

- \frac{1}{3} \cdot \frac{( 5 - 3 x ) ^{9}}{9} : - \frac{1}{27} (5 - 3 x)^{9}

= - \frac{1}{27} (5 - 3 x)^{9}

解答补常数

= - \frac{1}{27} (5 - 3 x)^{9} + C

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