Final answer:
To solve the given integral, we can use partial fraction decomposition. After decomposing the integrand and finding the values of the unknown coefficients, we can integrate each term separately and obtain the solution as ln|1-x| - 0.5ln(x²+1) - arctan(x) + C.
Step-by-step explanation:
To solve the integral ∫(3+x²)/(1-x)(x²+1) dx, we will use partial fraction decomposition. First, we factor the denominator as a product of linear and quadratic factors: (1-x)(x²+1). The quadratic factor cannot be factored further. Using partial fraction decomposition, we can express the integrand as a sum of two fractions: A/(1-x) + (Bx+C)/(x²+1).
Next, we find the values of A, B, and C by equating coefficients. After simplifying, we get A = 1, B = -1, and C = 1. Now, we can rewrite the integral as ∫(1/(1-x) - (x-1)/(x²+1)) dx.
Taking the integral of each term separately, we get ln|1-x| - 0.5ln(x²+1) - arctan(x) + C, where C is the constant of integration. Therefore, the solution to the integral is ln|1-x| - 0.5ln(x²+1) - arctan(x) + C.