63–68. Definite integrals Evaluate the following definite integrals. Use Theorem 7.7 to express your answer in terms of logarithms.
∫₋₂² dt/(t² – 9)

Verified step by step guidance

1. Identify the integral to evaluate: \(\displaystyle \int_{-2}^{2} \frac{dt}{t^{2} - 9}\).
2. Recognize that the denominator can be factored using the difference of squares: \(t^{2} - 9 = (t - 3)(t + 3)\).
3. Set up the partial fraction decomposition: \(\frac{1}{t^{2} - 9} = \frac{A}{t - 3} + \frac{B}{t + 3}\), and solve for constants \(A\) and \(B\).
4. Rewrite the integral as the sum of two integrals involving \(\frac{1}{t - 3}\) and \(\frac{1}{t + 3}\), then integrate each term separately to get expressions involving logarithms.
5. Apply the Fundamental Theorem of Calculus (Theorem 7.7) by evaluating the antiderivative at the upper limit \$2\( and the lower limit \)-2$, then subtract to find the definite integral.

Key Concepts

Definite Integrals

A definite integral calculates the net area under a curve between two limits. It is represented as ∫_a^b f(t) dt, where a and b are the bounds. Evaluating definite integrals often involves finding an antiderivative and then applying the Fundamental Theorem of Calculus.

Partial Fraction Decomposition

Partial fraction decomposition breaks a rational function into simpler fractions that are easier to integrate. For integrals involving rational functions like 1/(t² - 9), this method helps rewrite the integrand into terms whose antiderivatives are known, such as logarithmic functions.

Logarithmic Integration and Theorem 7.7

Theorem 7.7 typically refers to the integral of 1/(x - a) being ln|x - a| + C. When integrating rational functions that factor into linear terms, the antiderivative often involves logarithms. Expressing the answer in terms of logarithms means using this theorem to write the integral in a simplified logarithmic form.