Intergration help :)

integrate:

∫ dx/ [4 + x^(2)]

thanks alot if you can help :) if possible please leave some explaination :))

This form of integrals are handled by trigonometric substitutions.

Remembering the identity 1 + tan^2 x = sec^2 x, we make the substitution x = 2 tan u.
dx = 2 sec^2 u du
Our integral becomes:
∫ 1/(4+(2 tan u)^2 . 2sec^2u du
= ∫ 1/(4(1+tan^2 u)) . 2sec^2u du
= ∫ 2sec^2u / 4sec^2u du
= ∫ 2/4 du = ∫ 1/2 du
= u/2 + C

Now, we need to express u in terms of x.
x = 2 tan u
x/2 = tan u
arctan(x/2) = u

Substituting:

arctan(x/2)/2 + C is our final answer.

= (1/2) tan^-1 (x/2) + c