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

解答

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

\frac{1}{4 \cdot 2 ^{\frac{1}{3}}} (3 x + 10)^{\frac{4}{3}} + C

求解步骤

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

使用换元积分法

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

提出常数:

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

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

使用幂法则

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

u = \frac{3}{2} x + 5

代回

= \frac{2}{3} \cdot \frac{3}{4} (\frac{3}{2} x + 5)^{\frac{4}{3}}

化简

\frac{2}{3} \cdot \frac{3}{4} (\frac{3}{2} x + 5)^{\frac{4}{3}} : \frac{1}{4 \cdot 2 ^{\frac{1}{3}}} (3 x + 10)^{\frac{4}{3}}

= \frac{1}{4 \cdot 2 ^{\frac{1}{3}}} (3 x + 10)^{\frac{4}{3}}

解答补常数

= \frac{1}{4 \cdot 2 ^{\frac{1}{3}}} (3 x + 10)^{\frac{4}{3}} + C

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