Calculus: Early Transcendentals, Chapter 5, 5.5, Section 5.5, Problem 45

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