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

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

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

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

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

Solution steps

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

Apply u-substitution

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

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

Apply Trigonometric Substitution

: \int sec (v) d v

= \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

\frac{d}{d x} (\frac{x}{x - 10})

\frac{d y}{d x} = \frac{2 x - 3}{5 y}

\frac{d}{d x} (\frac{( x ^{3} - x )}{1 + x ^{2}})

\frac{d ^{2}}{d x ^{2}} (x e^{x})

\frac{d}{d x} (ln (\frac{( 1 - x )}{( x + 5 ) ^{3}}))

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