integral of (2e^x+4)/(2e^x-4)

解答

\int \frac{2 e ^{x} + 4}{2 e ^{x} - 4} d x

- x + 2 ln ∣ e^{x} - 2 ∣ + C

求解步骤

\int \frac{2 e ^{x} + 4}{2 e ^{x} - 4} d x

化简

\frac{2 e ^{x} + 4}{2 e ^{x} - 4} : \frac{e ^{x} + 2}{e ^{x} - 2}

= \int \frac{e ^{x} + 2}{e ^{x} - 2} d x

乘开

\frac{e ^{x} + 2}{e ^{x} - 2} : \frac{e ^{x}}{e ^{x} - 2} + \frac{2}{e ^{x} - 2}

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

= \int \frac{e ^{x}}{e ^{x} - 2} d x + \int \frac{2}{e ^{x} - 2} d x

\int \frac{e ^{x}}{e ^{x} - 2} d x = ln ∣ e^{x} - 2 ∣

\int \frac{2}{e ^{x} - 2} d x = - x + ln ∣ e^{x} - 2 ∣

= ln ∣ e^{x} - 2 ∣ - x + ln ∣ e^{x} - 2 ∣

化简

= - x + 2 ln ∣ e^{x} - 2 ∣

解答补常数

= - x + 2 ln ∣ e^{x} - 2 ∣ + C

\int \frac{1}{ln ( n )}

x \to + 1 + lim (\frac{2}{x} - \frac{1}{ln ( x )})

\frac{\partial}{\partial x} (x^{2} - y^{2} + z^{2} - 2 z - 4)

\int - 17 sin^{4} (x) cos (x) d x

d er i v a t i v e 5 x^{\frac{1}{3}} - 5 x^{- \frac{2}{3}}