integral of 1/((1-5x)^2)

解答

\int \frac{1}{( 1 - 5 x ) ^{2}} d x

\frac{1}{5 ( 1 - 5 x )} + C

求解步骤

\int \frac{1}{( 1 - 5 x ) ^{2}} d x

使用换元积分法

= \int - \frac{1}{5 u ^{2}} 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 \frac{1}{u ^{2}} d u

使用幂法则

= - \frac{1}{5} (- \frac{1}{u})

u = 1 - 5 x

代回

= - \frac{1}{5} (- \frac{1}{1 - 5 x})

化简

- \frac{1}{5} (- \frac{1}{1 - 5 x}) : \frac{1}{5 ( 1 - 5 x )}

= \frac{1}{5 ( 1 - 5 x )}

解答补常数

= \frac{1}{5 ( 1 - 5 x )} + C