Partial fractions integral with a cubic denominator using factoring by grouping.

We integrate (5x^2+11x+9)/(x^3-3x^2+x-3) using partial fractions decomposition to break the integrand into simpler pieces.

First, we note that the degree of the numerator is less than the degree of the denominator, so we don't have to do polynomial long division to put the fraction in its proper form. We also note that the denominator is a cubic polynomial, and this needs to be factored in order to apply partial fractions decomposition to the integral. Fortunately this cubic polynomial factors quickly by grouping the first and second pair of terms.

Next, we make our partial fractions proposal. Since our denominator has one linear factor and one irreducible quadratic factor, we propose the decomposition A/(x-3)+(Bx+C)/(x^2+1). We clear all the denominators and gather the coefficients of powers of x in descending order.

Setting the coefficients of the powers of x equal on the left and right hand sides, we generate a system of three equations for A, B and C, which we solve using a combination of elimination and substitution.

Now that our undetermined coefficients are in fact determined, we break the integrand into simpler pieces and integrate each piece to get the final answer of the partial fractions integral.