`int (sec(2x) + tan(2x)) dx` Find the indefinite integral.

`int(sec(2x)+tan(2x))dx=`

Use additivity of integral: `int (f(x)+g(x))dx=int f(x)dx+int g(x)dx.` `int sec(2x)dx+int tan(2x)dx=`

Make the same substitution for both integrals: `u=2x,` `du=2dx=>dx=(du)/2`

`1/2int sec u du+1/2int tan u du=`

Now we have table integrals.

`1/2ln|sec u+tan u|-1/2ln|cos u|+C`

Return the substitution to obtain the final result.

`1/2ln|sec(2x)+tan(2x)|-1/2ln|cos(2x)|+C` 

`int(sec(2x)+tan(2x))dx=`

Use additivity of integral: `int (f(x)+g(x))dx=int f(x)dx+int g(x)dx.` `int sec(2x)dx+int tan(2x)dx=`

Make the same substitution for both integrals: `u=2x,` `du=2dx=>dx=(du)/2`

`1/2int sec u du+1/2int tan u du=`

Now we have table integrals.

`1/2ln|sec u+tan u|-1/2ln|cos u|+C`

Return the substitution to obtain the final result.

`1/2ln|sec(2x)+tan(2x)|-1/2ln|cos(2x)|+C`