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受欢迎的微积分 >

integral of (3e^x)/(e^{2x)+2e^x+1}

解答

\int \frac{3 e ^{x}}{e ^{2 x} + 2 e ^{x} + 1} d x

解答

- \frac{3}{e ^{x} + 1} + C

求解步骤

\int \frac{3 e ^{x}}{e ^{2 x} + 2 e ^{x} + 1} d x

提出常数:

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

= 3 \cdot \int \frac{e ^{x}}{e ^{2 x} + 2 e ^{x} + 1} d x

分解

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

= 3 \cdot \int \frac{e ^{x}}{( e ^{x} + 1 ) ^{2}} d x

使用换元积分法

= 3 \cdot \int \frac{1}{u ^{2}} d u

使用幂法则

= 3 (- \frac{1}{u})

u = e^{x} + 1

代回

= 3 (- \frac{1}{e ^{x} + 1})

化简

3 (- \frac{1}{e ^{x} + 1}) : - \frac{3}{e ^{x} + 1}

= - \frac{3}{e ^{x} + 1}

解答补常数

= - \frac{3}{e ^{x} + 1} + C

作图

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