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

解答

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

1 + e^{2 x} + C

求解步骤

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

使用换元积分法

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

提出常数:

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

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

使用幂法则

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

u = 1 + e^{2 x}

代回

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

化简

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

= 1 + e^{2 x}

解答补常数

= 1 + e^{2 x} + C

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