integral of (ln^3(1-x))/(1-x)

解答

\int \frac{ln ^{3} ( 1 - x )}{1 - x} d x

- \frac{ln ^{4} ( 1 - x )}{4} + C

求解步骤

\int \frac{ln ^{3} ( 1 - x )}{1 - x} d x

使用换元积分法

= \int - \frac{ln ^{3} ( u )}{u} d u

提出常数:

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

= - \int \frac{ln ^{3} ( u )}{u} d u

使用换元积分法

= - \int v^{3} d v

使用幂法则

= - \frac{v ^{4}}{4}

代回

= - \frac{ln ^{4} ( 1 - x )}{4}

解答补常数

= - \frac{ln ^{4} ( 1 - x )}{4} + C

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