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

The integral of (1+e^x)/(e^{2x)+4} is 1/8 (-ln|e^{2x}+4|+2x)+1/2 arctan((e^x)/2)+C

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

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

\frac{1}{8} (- ln e^{2 x} + 4 + 2 x) + \frac{1}{2} arctan (\frac{e ^{x}}{2}) + C

Solution steps

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

Expand

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

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

\int \frac{1}{e ^{2 x} + 4} d x = \frac{1}{8} (- ln e^{2 x} + 4 + 2 x)

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

= \frac{1}{8} (- ln e^{2 x} + 4 + 2 x) + \frac{1}{2} arctan (\frac{e ^{x}}{2})

Add a constant to the solution

= \frac{1}{8} (- ln e^{2 x} + 4 + 2 x) + \frac{1}{2} arctan (\frac{e ^{x}}{2}) + C

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What is the integral of (1+e^x)/(e^{2x)+4} ?
The integral of (1+e^x)/(e^{2x)+4} is 1/8 (-ln|e^{2x}+4|+2x)+1/2 arctan((e^x)/2)+C