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受欢迎的微积分 >

integral of (2x^2)/(sqrt(1+x^3))

解答

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

解答

\frac{4}{3} 1 + x^{3} + C

求解步骤

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

提出常数:

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

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

使用换元积分法

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

提出常数:

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

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

使用幂法则

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

u = 1 + x^{3}

代回

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

化简

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

= \frac{4}{3} 1 + x^{3}

解答补常数

= \frac{4}{3} 1 + x^{3} + C

作图

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