What is the integral of (4e^{2x}+2e^x)/(1+e^{2x)} ?

The integral of (4e^{2x}+2e^x)/(1+e^{2x)} is 2ln|1+e^{2x}|+2arctan(e^x)+C

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integral of (4e^{2x}+2e^x)/(1+e^{2x)}

\int \frac{4 e ^{2 x} + 2 e ^{x}}{1 + e ^{2 x}} d x

2 ln 1 + e^{2 x} + 2 arctan (e^{x}) + C

Solution steps

\int \frac{4 e ^{2 x} + 2 e ^{x}}{1 + e ^{2 x}} d x

Expand

\frac{4 e ^{2 x} + 2 e ^{x}}{1 + e ^{2 x}} : \frac{4 e ^{2 x}}{1 + e ^{2 x}} + \frac{2 e ^{x}}{1 + e ^{2 x}}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{4 e ^{2 x}}{1 + e ^{2 x}} d x + \int \frac{2 e ^{x}}{1 + e ^{2 x}} d x

\int \frac{4 e ^{2 x}}{1 + e ^{2 x}} d x = 2 ln 1 + e^{2 x}

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

= 2 ln 1 + e^{2 x} + 2 arctan (e^{x})

Add a constant to the solution

= 2 ln 1 + e^{2 x} + 2 arctan (e^{x}) + C

\frac{d}{d x} (4 x - 3 x^{2})

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

\int x (20 x^{2} + 60 x) d x

\frac{5 + 5 s}{s ^{2} + 25}

x^{4} + 2 e^{x}

What is the integral of (4e^{2x}+2e^x)/(1+e^{2x)} ?
The integral of (4e^{2x}+2e^x)/(1+e^{2x)} is 2ln|1+e^{2x}|+2arctan(e^x)+C