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

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

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

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

arctan (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

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

Use the common integral:

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

= arctan (u)

Substitute back

u = e^{x}

= arctan (e^{x})

Add a constant to the solution

= arctan (e^{x}) + C

\int \frac{x ^{3} + 3 x ^{2} - 4}{x ^{2}} d x

\int \frac{2 x ^{3} + x ^{4} + x}{x ^{3}} d x

\int 3 x^{2} (x^{3} + 5)^{2} d x

\int coth (\frac{x}{5}) d x

\int 2 R e (z) d z

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