`int (1 + x)/(1 + x^2) dx` Evaluate the indefinite integral.

You need to evaluate the indefinite integral `int (1+x)/(1+x^2)dx` such that:

`int (1+x)/(1+x^2)dx = int 1/(1+x^2)dx + int x/(1+x^2)dx`

`int (1+x)/(1+x^2)dx = arctan x + int x/(1+x^2)dx`

You need to evaluate the indefinite integral` int x/(1+x^2)dx` using the following substitution  `x^2 + 1=u,` such that:

`x^2 + 1= u=>2xdx = du => xdx= (du)/2`

`int x/(1+x^2)dx = (1/2) int (du)/u`

`(1/2) int (du)/u = (1/2)*ln|u| + c`

Replacing back `x^2 + 1` for u yields:

`int x/(1+x^2)dx = (1/2) ln(x^2 + 1) + c`

Hence, evaluating the indefinite integral, yields `int (1+x)/(1+x^2)dx = arctan x + (1/2) ln(x^2 + 1) + c`