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受欢迎的微积分 >

integral of (15e^{5z})/(e^{5z)+2}

解答

\int \frac{15 e ^{5 z}}{e ^{5 z} + 2} d z

解答

3 ln e^{5 z} + 2 + C

求解步骤

\int \frac{15 e ^{5 z}}{e ^{5 z} + 2} d z

提出常数:

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

= 15 \cdot \int \frac{e ^{5 z}}{e ^{5 z} + 2} d z

使用换元积分法

= 15 \cdot \int \frac{1}{5 u} d u

提出常数:

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

= 15 \cdot \frac{1}{5} \cdot \int \frac{1}{u} d u

使用积分常见定则:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 15 \cdot \frac{1}{5} ln ∣ u ∣

u = e^{5 z} + 2

代回

= 15 \cdot \frac{1}{5} ln e^{5 z} + 2

化简

15 \cdot \frac{1}{5} ln e^{5 z} + 2 : 3 ln e^{5 z} + 2

= 3 ln e^{5 z} + 2

解答补常数

= 3 ln e^{5 z} + 2 + C

作图

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