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