integral of (e^{8x})/(e^{8x)+7}

解答

\int \frac{e ^{8 x}}{e ^{8 x} + 7} d x

\frac{1}{8} ln e^{8 x} + 7 + C

求解步骤

\int \frac{e ^{8 x}}{e ^{8 x} + 7} d x

使用换元积分法

= \int \frac{1}{8 u} d u

提出常数:

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

= \frac{1}{8} \cdot \int \frac{1}{u} d u

使用积分常见定则:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{8} ln ∣ u ∣

u = e^{8 x} + 7

代回

= \frac{1}{8} ln e^{8 x} + 7

解答补常数

= \frac{1}{8} ln e^{8 x} + 7 + C

x \to - \infty lim (x)

d er i v a t i v e y = x^{11} e^{x}

\frac{\partial}{\partial x} (- 7 x^{2} + 7 y^{2} - 7)

\frac{\partial}{\partial y} (\frac{x + y}{1 + y ^{2}})

\frac{d}{d x} (- e^{- 5 x})