Question

Evaluate. $$ \int_{1}^{2} \frac{x^{2}-x-6}{x+2} d x $$

   Evaluate.
$$
\int_{1}^{2} \frac{x^{2}-x-6}{x+2} d x
$$

Calculus

Earl W. Swokowski 5th Edition

Chapter 5, Problem 18

Instant Answer

Solved by Expert Nick Johnson

03/25/2022

Step 1

The numerator $x^{2}-x-6$ can be factored into $(x+2)(x-3)$. So, the integral becomes: $$ \int_{1}^{2} \frac{(x+2)(x-3)}{x+2} d x $$ Show more…

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Evaluate. $$ \int_{1}^{2} \frac{x^{2}-x-6}{x+2} d x $$

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Key Concepts

Polynomial Division

Polynomial division is a technique used to simplify a rational function when the degree of the numerator is greater than or equal to the degree of the denominator. By dividing the numerator by the denominator, the rational expression can often be rewritten as the sum of a polynomial and a simpler rational expression, which makes the process of integration easier.

Definite Integrals

Definite integrals calculate the accumulation of quantities, representing the net area under a curve between specified limits. When evaluating a definite integral, one first finds the antiderivative of the function and then uses the Fundamental Theorem of Calculus to compute the difference between the values of the antiderivative at the upper and lower limits.

Integration of Rational Functions

Integration of rational functions involves finding the antiderivative of a function expressed as a ratio of two polynomials. Techniques such as polynomial division and partial fraction decomposition are commonly employed to simplify the function into forms that are easier to integrate. This step is critical in evaluating integrals that arise in many areas of calculus.

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