Question

What is `int 2/(4+x^(2)) dx`?

Answer 1

The answer is `=arctan(x/2)+C`

Explanation:

Perform this integral by substitution

`4+x^2=4(1+(x/2)^2)`

Let `tanu=x/2`

`sec^2udu=1/2dx`

`1+tan^2u=sec^2u`

Therefore, the integral is

`int(2dx)/(4+x^2)=int(4sec^2udu)/(4(1+tan^2u))`

`=int(sec^2udu)/(sec^2u)`

`=int(du)`

`=u`

`=arctan(x/2)+C`

Answer 2

The integral is equal to `arctan(x/2)+C`.

Explanation:

To solve the integral, use the substitution `u=x/2`, which means `x=2u` and `dx=2du`:

`color(white)=int2/(4+x^2)` `dx`

`=int2/(4+(2u)^2)` `2du`

`=int4/(4+4u^2)` `du`

`=intcolor(red)cancelcolor(black)4/(color(red)cancelcolor(black)4(1+u^2))` `du`

`=int1/(1+u^2)` `du`

`=arctan(u)+C`

`=arctan(x/2)+C`

That's the integral. Hope this helped!