How to integrate (x^3+1)/(x^2+1)^2?

To integrate (x^3+1)/(x^2+1)^2, use partial fractions and substitution.

First, use partial fractions to break up the integrand into simpler fractions. Let (x^3+1)/(x^2+1)^2 = A/(x^2+1) + B/(x^2+1)^2. Then, cross-multiply and solve for A and B to get A = 1/2 and B = -1/2.

Next, substitute u = x^2+1 and du = 2x dx. Then, the integral becomes ∫(1/2)(1/u) du - ∫(1/2)(1/u^2) du. Integrating each term separately, we get (1/2)ln|x^2+1| + (1/2)(1/(x^2+1)) + C, where C is the constant of integration.

Therefore, the final answer is (1/2)ln|x^2+1| + (1/2)(1/(x^2+1)) + C.