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受欢迎的微积分 >

integral of (3x^2)/(\sqrt[3]{x^3+1)}

解答

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

解答

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

求解步骤

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

提出常数:

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

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

使用换元积分法

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

提出常数:

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

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

使用幂法则

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

u = x^{3} + 1

代回

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

化简

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

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

解答补常数

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

作图

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