integral of (e^x)/((1+5e^x)^{1/2)}

解答

\int \frac{e ^{x}}{( 1 + 5 e ^{x} ) ^{\frac{1}{2}}} d x

\frac{2}{5} (1 + 5 e^{x})^{\frac{1}{2}} + C

求解步骤

\int \frac{e ^{x}}{( 1 + 5 e ^{x} ) ^{\frac{1}{2}}} d x

使用换元积分法

= \int \frac{1}{5 u ^{\frac{1}{2}}} d u

提出常数:

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

= \frac{1}{5} \cdot \int \frac{1}{u ^{\frac{1}{2}}} d u

使用幂法则

= \frac{1}{5} \cdot 2 u^{\frac{1}{2}}

u = 1 + 5 e^{x}

代回

= \frac{1}{5} \cdot 2 (1 + 5 e^{x})^{\frac{1}{2}}

化简

\frac{1}{5} \cdot 2 (1 + 5 e^{x})^{\frac{1}{2}} : \frac{2}{5} (1 + 5 e^{x})^{\frac{1}{2}}

= \frac{2}{5} (1 + 5 e^{x})^{\frac{1}{2}}

解答补常数

= \frac{2}{5} (1 + 5 e^{x})^{\frac{1}{2}} + C

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