Evaluate the integral of 1/ ( 1 + 4x^2)

We have to
find Int [1/ (1 + 4x^2) dx].

First substitute u = 2x

=>
du /dx = 2

=> du /2 = dx

Now Int [1/ (1 + 4x^2)
dx]

=> Int [(1/2)*(1/ (1+u^2) du]

=> (1/2)*Int [1/
(1 + u^2) du]

Now Int [1/ (1+u^2) du] =arc tan u + C

=>
(1/2)*arc tan u + C

replace u with 2x

=> (1/2)* arc tan
2x + C

Therefore the required result is

1/2)* arc tan 2x +
C