integral of (\sqrt[3]{x^2})/3

解答

\int \frac{3 x ^{2}}{3} d x

\frac{1}{5} x^{\frac{5}{3}} + C

求解步骤

\int \frac{3 x ^{2}}{3} d x

提出常数:

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

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

3 x^{2} = x^{\frac{2}{3}},

假定

x \geq 0

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

使用幂法则

= \frac{1}{3} \cdot \frac{3}{5} x^{\frac{5}{3}}

化简

\frac{1}{3} \cdot \frac{3}{5} x^{\frac{5}{3}} : \frac{1}{5} x^{\frac{5}{3}}

= \frac{1}{5} x^{\frac{5}{3}}

解答补常数

= \frac{1}{5} x^{\frac{5}{3}} + C

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