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

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

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

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

\frac{1}{2} x + \frac{1}{2} ln ∣ e^{x} + 4 ∣ + C

Solution steps

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

Expand

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

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

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

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

Simplify

ln ∣ e^{x} + 4 ∣ + 2 (\frac{1}{4} x - \frac{1}{4} ln ∣ e^{x} + 4 ∣) : \frac{1}{2} x + \frac{1}{2} ln ∣ e^{x} + 4 ∣

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

Add a constant to the solution

= \frac{1}{2} x + \frac{1}{2} ln ∣ e^{x} + 4 ∣ + C

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