integral of 1/((5x+2)^3)

解答

\int \frac{1}{( 5 x + 2 ) ^{3}} d x

- \frac{1}{10 ( 5 x + 2 ) ^{2}} + C

求解步骤

\int \frac{1}{( 5 x + 2 ) ^{3}} d x

使用换元积分法

= \int \frac{1}{5 u ^{3}} 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 ^{3}} d u

使用幂法则

= \frac{1}{5} (- \frac{1}{2 u ^{2}})

u = 5 x + 2

代回

= \frac{1}{5} (- \frac{1}{2 ( 5 x + 2 ) ^{2}})

化简

\frac{1}{5} (- \frac{1}{2 ( 5 x + 2 ) ^{2}}) : - \frac{1}{10 ( 5 x + 2 ) ^{2}}

= - \frac{1}{10 ( 5 x + 2 ) ^{2}}

解答补常数

= - \frac{1}{10 ( 5 x + 2 ) ^{2}} + C

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