I'm going to assume that you meant to use parenthesis around portions of your derivative function and that it is supposed to be:

That means to find the function f(x) that we must integrate:

Let's first break this fraction (and the associated integral) into two:

f(x) = ∫1/(x^{2}+1)dx - ∫(2x)/(x^{2}+1)dx

To make it easier to explain, let's call the problem f(x) = A - B where A = ∫1/(x^{2}+1)dx and B = ∫(2x)/(x^{2}+1)dx

Integrating A is easy if you remember that the derivative of arctan(x) = 1/(x^{2} + 1) which means that the integral of 1/(x^{2} + 1)dx is arctan(x).

To Integrate B, let u = x^{2} + 1, the du/dx = 2x so du = 2xdx meaning that we can write the integral as:

∫1/u du and the integral of this is ln(u). Back-substituting, we get ln(x^{2} + 1).

So f(x) = A - B becomes:

f(x) = arctan(x) - ln(x^{2} + 1) + C

To find the value of C, we can plug in the point (0, 6) so:

6 = arctan(0) - ln(0^{2} + 1) + C

6 = 0 - 0 + C

C = 6

f(x) = arctan(x) - ln(x^{2} + 1) + 6