integral from-1 to 1 of ((3e^x))/(1+e^x)

解答

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

3

求解步骤

\int_{- 1}^{1} \frac{3 e ^{x}}{1 + e ^{x}} d x

提出常数:

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

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

使用换元积分法

= 3 \cdot \int_{1 + \frac{1}{e}}^{1 + e} \frac{1}{u} d u

使用积分常见定则:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 3 [ln ∣ u ∣]_{1 + \frac{1}{e}}^{1 + e}

带入上下限计算后得:

ln (1 + e) - ln (1 + \frac{1}{e})

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

化简

= 3

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