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

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

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

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

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

Solution steps

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

Apply u-substitution

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

Use the common integral:

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

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