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受欢迎的微积分 >

integral of 4e^xsqrt(8+e^x)

解答

\int 4 e^{x} 8 + e^{x} d x

解答

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

求解步骤

\int 4 e^{x} 8 + e^{x} d x

提出常数:

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

= 4 \cdot \int e^{x} 8 + e^{x} d x

使用换元积分法

= 4 \cdot \int u d u

使用幂法则

= 4 \cdot \frac{2}{3} u^{\frac{3}{2}}

u = 8 + e^{x}

代回

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

化简

4 \cdot \frac{2}{3} (8 + e^{x})^{\frac{3}{2}} : \frac{8}{3} (8 + e^{x})^{\frac{3}{2}}

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

解答补常数

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

作图

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