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...

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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**