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受欢迎的微积分 >

integral of 1/(e^{x/3)+e^{x/6}}

解答

\int \frac{1}{e ^{\frac{x}{3}} + e ^{\frac{x}{6}}} d x

解答

- x - \frac{6}{e ^{\frac{x}{6}}} + 6 ln e^{\frac{x}{6}} + 1 + C

求解步骤

\int \frac{1}{e ^{\frac{x}{3}} + e ^{\frac{x}{6}}} d x

使用换元积分法

= \int \frac{3}{e ^{u} + e ^{\frac{u}{2}}} d u

提出常数:

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

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

使用换元积分法

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

提出常数:

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

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

使用换元积分法

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

将

\frac{1}{w ( w ^{2} + w )}

用部份分式展开:

- \frac{1}{w} + \frac{1}{w ^{2}} + \frac{1}{w + 1}

= 3 \cdot 2 \cdot \int - \frac{1}{w} + \frac{1}{w ^{2}} + \frac{1}{w + 1} d w

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

= 3 \cdot 2 (- \int \frac{1}{w} d w + \int \frac{1}{w ^{2}} d w + \int \frac{1}{w + 1} d w)

\int \frac{1}{w} d w = ln ∣ w ∣

\int \frac{1}{w ^{2}} d w = - \frac{1}{w}

\int \frac{1}{w + 1} d w = ln ∣ w + 1 ∣

= 3 \cdot 2 (- ln ∣ w ∣ - \frac{1}{w} + ln ∣ w + 1 ∣)

代回

= 3 \cdot 2 (- ln e^{\frac{\frac{x}{3}}{2}} - \frac{1}{e ^{\frac{\frac{x}{3}}{2}}} + ln e^{\frac{\frac{x}{3}}{2}} + 1)

化简

3 \cdot 2 (- ln e^{\frac{\frac{x}{3}}{2}} - \frac{1}{e ^{\frac{\frac{x}{3}}{2}}} + ln e^{\frac{\frac{x}{3}}{2}} + 1) : - x - \frac{6}{e ^{\frac{x}{6}}} + 6 ln e^{\frac{x}{6}} + 1

= - x - \frac{6}{e ^{\frac{x}{6}}} + 6 ln e^{\frac{x}{6}} + 1

解答补常数

= - x - \frac{6}{e ^{\frac{x}{6}}} + 6 ln e^{\frac{x}{6}} + 1 + C

作图

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