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

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

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

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

ln e^{2 x} + 1 + e^{x} + C

Solution steps

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

Apply u-substitution

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

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

Use the common integral:

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

= ln u^{2} + 1 + u

Substitute back

u = e^{x}

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

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= ln e^{2 x} + 1 + e^{x}

Add a constant to the solution

= ln e^{2 x} + 1 + e^{x} + C

\int_{2}^{3} \frac{5}{3 - x} d x

\frac{\partial}{\partial x} (x^{2} \cdot y^{3})

\int x (x + a) (x + b) d x

y^{^{''}} + 20 y^{^{'}} + 500 = 12

\int \frac{1}{x - 1 + 1 + x} d x

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