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
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