integral of 6/((x+5)^3)

解答

\int \frac{6}{( x + 5 ) ^{3}} d x

- \frac{3}{( x + 5 ) ^{2}} + C

求解步骤

\int \frac{6}{( x + 5 ) ^{3}} d x

提出常数:

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

= 6 \cdot \int \frac{1}{( x + 5 ) ^{3}} d x

使用换元积分法

= 6 \cdot \int \frac{1}{u ^{3}} d u

使用幂法则

= 6 (- \frac{1}{2 u ^{2}})

u = x + 5

代回

= 6 (- \frac{1}{2 ( x + 5 ) ^{2}})

化简

6 (- \frac{1}{2 ( x + 5 ) ^{2}}) : - \frac{3}{( x + 5 ) ^{2}}

= - \frac{3}{( x + 5 ) ^{2}}

解答补常数

= - \frac{3}{( x + 5 ) ^{2}} + C

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