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

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

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

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

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

Solution steps

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

Apply u-substitution

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

Apply Trigonometric Substitution

= \int sec (v) d v

Use the common integral:

\int sec (v) d v = ln ∣ tan (v) + sec (v) ∣

= ln ∣ tan (v) + sec (v) ∣

Substitute back

= ln tan (arctan (\frac{1}{2} e^{x})) + sec (arctan (\frac{1}{2} e^{x}))

Simplify

ln tan (arctan (\frac{1}{2} e^{x})) + sec (arctan (\frac{1}{2} e^{x})) : ln (\frac{1}{2} e^{x} + 4 + e^{2 x})

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

Add a constant to the solution

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

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