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受欢迎的微积分 >

integral of 4x(x^2+25000)^{-2/3}

解答

\int 4 x (x^{2} + 25000)^{- \frac{2}{3}} d x

解答

6 (x^{2} + 25000)^{\frac{1}{3}} + C

求解步骤

\int 4 x (x^{2} + 25000)^{- \frac{2}{3}} d x

提出常数:

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

= 4 \cdot \int x (x^{2} + 25000)^{- \frac{2}{3}} d x

使用换元积分法

= 4 \cdot \int \frac{1}{2 u ^{\frac{2}{3}}} d u

提出常数:

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

= 4 \cdot \frac{1}{2} \cdot \int \frac{1}{u ^{\frac{2}{3}}} d u

使用幂法则

= 4 \cdot \frac{1}{2} \cdot 3 u^{\frac{1}{3}}

u = x^{2} + 25000

代回

= 4 \cdot \frac{1}{2} \cdot 3 (x^{2} + 25000)^{\frac{1}{3}}

化简

4 \cdot \frac{1}{2} \cdot 3 (x^{2} + 25000)^{\frac{1}{3}} : 6 (x^{2} + 25000)^{\frac{1}{3}}

= 6 (x^{2} + 25000)^{\frac{1}{3}}

解答补常数

= 6 (x^{2} + 25000)^{\frac{1}{3}} + C

作图

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