# Complete the following exercise: int_2^3 {x^2 + 3x}/{x^3 + x^2 - x - 1} dx = int_2^3 ({}/{(x +...

## Question:

Complete the following exercise:

{eq}\begin{align*} \qquad \int_2^3 \dfrac{x^2 + 3x}{x^3 + x^2 - x - 1}\ dx &= \int_2^3 \left( \dfrac{}{(x + 1)^2} + \dfrac{}{x - 1} \right)\ dx \\ &= \cdots \end{align*}\, {/eq}

## Integration by Partial Fractions:

When integrating a rational function, {eq}f(x) = \dfrac{p(x)}{q(x)} {/eq}, where {eq}p {/eq} and {eq}q {/eq} are polynomials and the degree of the denominator is greater than the degree of the numerator, the integrand is often rewritten using partial fractions to rewrite as a sum or difference of simpler integrals. To find the partial fraction expansion of {eq}f(x) = \dfrac{p(x)}{q(x)} {/eq}, begin by fully factoring the denominator. Then, each linear factor {eq}(ax+b) {/eq} of the denominator will correspond to a term {eq}\dfrac{A}{ax+b} {/eq} in the partial fraction expansion. If a linear factor appears multiple times, for example {eq}(ax+b)^n {/eq}, then there are {eq}n {/eq} corresponding terms in the partial fraction expansion: {eq}\dfrac{A_1}{ax+b} + \dfrac{A_2}{(ax+b)^2} + \dfrac{A_3}{(ax+b)^3} + \cdots + \dfrac{A_{n-1}}{(ax+b)^{n-1}} + \dfrac{A_n}{(ax+b)^n} {/eq}. If the denominator has an irreducible quadratic factor, {eq}ax^2 + bx + c {/eq}, then the partial fraction expansion has a corresponding term {eq}\dfrac{Ax+B}{ax^2+bx+c} {/eq}. If an irreducible quadratic factor appears multiple times, for example {eq}(ax^2+bx + c)^n {/eq}, then there are {eq}n {/eq} corresponding terms in the partial fraction expansion: {eq}\dfrac{A_1x + B_1}{ax^2 +bx + c} + \dfrac{A_2x + B_2}{(ax^2+bx + c)^2} + \dfrac{A_3x + B_3}{(ax^2+bx + c)^3} + \cdots + \dfrac{A_{n-1}x + B_{n-1}}{(ax^2+bx + c)^{n-1}} + \dfrac{A_nx + B_n}{(ax^2+bx + c)^n} {/eq}. All of the constants in the partial fraction expansion can be found using algebra, including using systems of equations.