Question

Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.∫ [(2x³ + x² − 21x + 24) / (x² + 2x − 8)] dx

Accepted Answer

First, observe the integral $$ \int \frac{2x^3 + x^2 - 21x + 24}{x^2 + 2x - 8} \, dx . $$Since the degree of the numerator (3) is higher than the degree of the denominator (2), start by performing polynomial long division to simplify the integrand. Divide $$ 2x^3 + x^2 - 21x + 24  by$$ $$ x^2 + 2x - 8  to $$express the integrand as a polynomial plus a proper rational function (where the numerator degree is less than the denominator degree). After the division, rewrite the integral as $$ \int (	ext{quotient} + \frac{	ext{remainder}}{x^2 + 2x - 8}) \, dx . $$Then, split the integral into two parts: one involving the polynomial quotient and the other involving the proper rational function. Next, factor the denominator $$ x^2 + 2x - 8 $$ into $$ (x + 4)(x - 2)  to $$prepare for partial fraction decomposition of the proper rational function. Set up the partial fraction decomposition for the proper rational function as $$ \frac{A}{x + 4} + \frac{B}{x - 2} $$, solve for constants $$ A $$ and $$ B $$, then integrate each term separately.

Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [(2x³ + x² − 21x + 24) / (x² + 2x − 8)] dx

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Key Concepts

Polynomial Long Division

Substitution Method

Partial Fraction Decomposition