integral of ((e^{2x}+2e^x+1)/(1+e^x))

解答

\int (\frac{e ^{2 x} + 2 e ^{x} + 1}{1 + e ^{x}}) d x

1 + e^{x} + x + C

求解步骤

\int \frac{e ^{2 x} + 2 e ^{x} + 1}{1 + e ^{x}} d x

乘开

\frac{e ^{2 x} + 2 e ^{x} + 1}{1 + e ^{x}} : \frac{e ^{2 x}}{1 + e ^{x}} + \frac{2 e ^{x}}{1 + e ^{x}} + \frac{1}{1 + e ^{x}}

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 ^{2 x}}{1 + e ^{x}} d x + \int \frac{2 e ^{x}}{1 + e ^{x}} d x + \int \frac{1}{1 + e ^{x}} d x

\int \frac{e ^{2 x}}{1 + e ^{x}} d x = 1 + e^{x} - ln ∣ 1 + e^{x} ∣

\int \frac{2 e ^{x}}{1 + e ^{x}} d x = 2 ln ∣ 1 + e^{x} ∣

\int \frac{1}{1 + e ^{x}} d x = - ln ∣ 1 + e^{x} ∣ + x

= 1 + e^{x} - ln ∣ 1 + e^{x} ∣ + 2 ln ∣ 1 + e^{x} ∣ - ln ∣ 1 + e^{x} ∣ + x

化简

= 1 + e^{x} + x

解答补常数

= 1 + e^{x} + x + C

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