Integrate1 plus x divided by x2 plus 1?

Answer 1

(1+x)/(x^2+1)

Let x^2+1 =u

2x dx = du

x dx = du/2

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

Integral of x dx / (x^2+1) = (1/2) integral du / u = 1/2 ln|u| --(1)

Integral of 1 / (x^2+1) = arctan(x) --(2)

Adding (1) and (2)

Integral (1+x)/(x^2+1) = (1/2) ln(x^2+1) + arctan(x) + C