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受欢迎的微积分 >

integral of 7u^7sqrt(u^8+7)

解答

\int 7 u^{7} u^{8} + 7 d u

解答

\frac{7}{12} (u^{8} + 7)^{\frac{3}{2}} + C

求解步骤

\int 7 u^{7} u^{8} + 7 d u

提出常数:

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

= 7 \cdot \int u^{7} u^{8} + 7 d u

使用换元积分法

= 7 \cdot \int \frac{v}{8} d v

提出常数:

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

= 7 \cdot \frac{1}{8} \cdot \int v d v

使用幂法则

= 7 \cdot \frac{1}{8} \cdot \frac{2}{3} v^{\frac{3}{2}}

v = u^{8} + 7

代回

= 7 \cdot \frac{1}{8} \cdot \frac{2}{3} (u^{8} + 7)^{\frac{3}{2}}

化简

7 \cdot \frac{1}{8} \cdot \frac{2}{3} (u^{8} + 7)^{\frac{3}{2}} : \frac{7}{12} (u^{8} + 7)^{\frac{3}{2}}

= \frac{7}{12} (u^{8} + 7)^{\frac{3}{2}}

解答补常数

= \frac{7}{12} (u^{8} + 7)^{\frac{3}{2}} + C

作图

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