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受欢迎的微积分 >

integral of (e^x)/(e^x(e^x-3))

解答

\int \frac{e ^{x}}{e ^{x} ( e ^{x} - 3 )} d x

解答

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

求解步骤

\int \frac{e ^{x}}{e ^{x} ( e ^{x} - 3 )} d x

约分:

e^{x}

= \int \frac{1}{e ^{x} - 3} d x

使用换元积分法

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

将

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

用部份分式展开:

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

= \int - \frac{1}{3 u} + \frac{1}{3 ( u - 3 )} d u

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int \frac{1}{3 u} d u + \int \frac{1}{3 ( u - 3 )} d u

\int \frac{1}{3 u} d u = \frac{1}{3} ln ∣ u ∣

\int \frac{1}{3 ( u - 3 )} d u = \frac{1}{3} ln ∣ u - 3 ∣

= - \frac{1}{3} ln ∣ u ∣ + \frac{1}{3} ln ∣ u - 3 ∣

u = e^{x}

代回

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

\frac{1}{3} ln ∣ e^{x} ∣ = \frac{1}{3} x

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

解答补常数

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

作图

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