`int_0^ln5 e^x/(1+e^(2x)) dx` Evaluate the definite integral

For the given integral problem:` int_0^(ln(5))e^x/(1+e^(2x))dx` , it resembles the basic integration formula for inverse tangent:

`int_a^b (du)/(u^2+c^2) = (1/c)arctan(u/c) |_a^b`

where we let:

`u^2 =e^(2x) ` or` (e^x)^2 `   then `u= e^x`

`c^2 =1` or `1^2` then `c=1`

For the derivative of `u =e^(x)` , we apply the derivative of exponential function:

`du =e^x dx` .

Applying u-substitution: `u = e^x ` and` du = e^x dx` , we get:

`int e^x/(1+e^(2x))dx =int (e^xdx)/(1+(e^x)^2)`

                              `=int (du)/(1+(u)^2)`

Applying the basic integral formula of inverse tangent, we get:

`int (du)/(1+(u)^2) =(1/1)arctan(u/1)`

                           = `arctan(u)`

Express it in terms of x by plug-in `u=e^x` :

`arctan(u) =arctan(e^x)`

Evaluate with the given boundary limit:

`arctan(e^x)|_0^(ln(5)) =arctan(e^(ln(5)))-arctan(e^0)`

                          ` =arctan(5)-arctan(1)`

                                ` =arctan(5) -pi/4`