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受欢迎的微积分 >

integral of (2t+3)^{-3}

解答

\int (2 t + 3)^{- 3} d t

解答

- \frac{1}{4 ( 2 t + 3 ) ^{2}} + C

求解步骤

\int (2 t + 3)^{- 3} d t

使用指数法则:

a^{- b} = \frac{1}{a ^{b}}

= \int \frac{1}{( 2 t + 3 ) ^{3}} d t

使用换元积分法

= \int \frac{1}{2 u ^{3}} d u

提出常数:

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

= \frac{1}{2} \cdot \int \frac{1}{u ^{3}} d u

使用幂法则

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

u = 2 t + 3

代回

= \frac{1}{2} (- \frac{1}{2 ( 2 t + 3 ) ^{2}})

化简

\frac{1}{2} (- \frac{1}{2 ( 2 t + 3 ) ^{2}}) : - \frac{1}{4 ( 2 t + 3 ) ^{2}}

= - \frac{1}{4 ( 2 t + 3 ) ^{2}}

解答补常数

= - \frac{1}{4 ( 2 t + 3 ) ^{2}} + C

作图

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