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受欢迎的微积分 >

integral of 10x^2\sqrt[4]{8+2x^3}

解答

\int 10 x^{2} 4 8 + 2 x^{3} d x

解答

\frac{4}{3} (2 x^{3} + 8)^{\frac{5}{4}} + C

求解步骤

\int 10 x^{2} 4 8 + 2 x^{3} d x

提出常数:

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

= 10 \cdot \int x^{2} 4 8 + 2 x^{3} d x

使用换元积分法

= 10 \cdot \int \frac{4 u}{6} d u

提出常数:

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

= 10 \cdot \frac{1}{6} \cdot \int 4 u d u

使用幂法则

= 10 \cdot \frac{1}{6} \cdot \frac{4}{5} u^{\frac{5}{4}}

u = 8 + 2 x^{3}

代回

= 10 \cdot \frac{1}{6} \cdot \frac{4}{5} (8 + 2 x^{3})^{\frac{5}{4}}

化简

10 \cdot \frac{1}{6} \cdot \frac{4}{5} (8 + 2 x^{3})^{\frac{5}{4}} : \frac{4}{3} (2 x^{3} + 8)^{\frac{5}{4}}

= \frac{4}{3} (2 x^{3} + 8)^{\frac{5}{4}}

解答补常数

= \frac{4}{3} (2 x^{3} + 8)^{\frac{5}{4}} + C

作图

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