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受欢迎的微积分 >

integral of 4/(4+e^x)

解答

\int \frac{4}{4 + e ^{x}} d x

解答

x - ln ∣ e^{x} + 4 ∣ + C

求解步骤

\int \frac{4}{4 + e ^{x}} d x

提出常数:

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

= 4 \cdot \int \frac{1}{4 + e ^{x}} d x

使用换元积分法

= 4 \cdot \int \frac{1}{u ( 4 + u )} d u

将

\frac{1}{u ( 4 + u )}

用部份分式展开:

\frac{1}{4 u} - \frac{1}{4 ( u + 4 )}

= 4 \cdot \int \frac{1}{4 u} - \frac{1}{4 ( u + 4 )} 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

= 4 (\int \frac{1}{4 u} d u - \int \frac{1}{4 ( u + 4 )} d u)

\int \frac{1}{4 u} d u = \frac{1}{4} ln ∣ u ∣

\int \frac{1}{4 ( u + 4 )} d u = \frac{1}{4} ln ∣ u + 4 ∣

= 4 (\frac{1}{4} ln ∣ u ∣ - \frac{1}{4} ln ∣ u + 4 ∣)

u = e^{x}

代回

= 4 (\frac{1}{4} ln ∣ e^{x} ∣ - \frac{1}{4} ln ∣ e^{x} + 4 ∣)

化简

4 (\frac{1}{4} ln ∣ e^{x} ∣ - \frac{1}{4} ln ∣ e^{x} + 4 ∣) : x - ln ∣ e^{x} + 4 ∣

= x - ln ∣ e^{x} + 4 ∣

解答补常数

= x - ln ∣ e^{x} + 4 ∣ + C

作图

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