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受欢迎的微积分 >

integral of 5/(3e^x-2)

解答

\int \frac{5}{3 e ^{x} - 2} d x

解答

5 (\frac{1}{2} ln ∣ 3 e^{x} - 2 ∣ - \frac{1}{2} (ln (3) + x)) + C

求解步骤

\int \frac{5}{3 e ^{x} - 2} d x

提出常数:

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

= 5 \cdot \int \frac{1}{3 e ^{x} - 2} d x

使用换元积分法

= 5 \cdot \int \frac{1}{u ( u + 2 )} d u

将

\frac{1}{u ( u + 2 )}

用部份分式展开:

\frac{1}{2 u} - \frac{1}{2 ( u + 2 )}

= 5 \cdot \int \frac{1}{2 u} - \frac{1}{2 ( u + 2 )} 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

= 5 (\int \frac{1}{2 u} d u - \int \frac{1}{2 ( u + 2 )} d u)

\int \frac{1}{2 u} d u = \frac{1}{2} ln ∣ u ∣

\int \frac{1}{2 ( u + 2 )} d u = \frac{1}{2} ln ∣ u + 2 ∣

= 5 (\frac{1}{2} ln ∣ u ∣ - \frac{1}{2} ln ∣ u + 2 ∣)

u = 3 e^{x} - 2

代回

= 5 (\frac{1}{2} ln ∣ 3 e^{x} - 2 ∣ - \frac{1}{2} ln ∣ 3 e^{x} - 2 + 2 ∣)

化简

5 (\frac{1}{2} ln ∣ 3 e^{x} - 2 ∣ - \frac{1}{2} ln ∣ 3 e^{x} - 2 + 2 ∣) : 5 (\frac{1}{2} ln ∣ 3 e^{x} - 2 ∣ - \frac{1}{2} (ln (3) + x))

= 5 (\frac{1}{2} ln ∣ 3 e^{x} - 2 ∣ - \frac{1}{2} (ln (3) + x))

解答补常数

= 5 (\frac{1}{2} ln ∣ 3 e^{x} - 2 ∣ - \frac{1}{2} (ln (3) + x)) + C

作图

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