Problem 31 - College Algebra and Calculus: An Applied Approach

Question

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. $$ \int_{3}^{4} \frac{1}{\sqrt{x^{2}-9}} d x $$

Step-by-Step Solution

Answer

The improper integral \(\int_{3}^{4} \frac{1}{\sqrt{x^{2}-9}} d x \) converges and has the value \(asinh(\frac{4}{3}) - asinh(1)\)

Step 1: Identify the Indefinite Integral

To solve this problem, it's necessary to identify the indefinite integral of \(\frac{1}{\sqrt{x^{2}-9}}\), which is \(asinh(\frac{x}{3})\). Using the property of indefinite integral \( \int f(x)dx = F(x) + C\), where F(x) is the antiderivative of f(x) and C is the constant of integration.

Step 2: Apply the Fundamental Theorem of Calculus.

Once you have the indefinite integral, you can apply the Fundamental Theorem of Calculus, which states that the definite integral of a function can be determined by evaluating the difference in its antiderivative at the end points of the interval [a, b]. In this case, it means evaluating \(asinh(\frac{x}{3})\) at x = 4 and x = 3.

Step 3: Evaluate and Subtract

Evaluate the antiderivative at each point: \(asinh(\frac{4}{3}) - asinh(\frac{3}{3})\). Subtract the evaluated antiderivative at the lower limit of integration from the evaluated antiderivative at the upper limit. This will give the numerical solution for the definite integral.

Step 4: Check if Integral Converges or Diverges

An improper integral converges if the limit as the upper/lower bound approaches infinity or a point where the integrand is undefined is a finite number. Since our integrand is finite over the entire interval [3,4], excluding the endpoints, and we've computed a numerical value for the integral, we can conclude that the improper integral converges.