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Chapter 6: Problem 36

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. $$ \int_{0}^{2} \frac{1}{4-x^{2}} d x $$

Short Answer

Expert verified
The improper integral converges and the evaluated result is \( \frac{\pi}{4} \).

Step by step solution

01

Identify the singularity

Identify that the singularity of the function \( f(x) = \frac{1}{4-x^{2}} \) is at x = 2. This makes the integral improper.
02

Rewrite the integral

Rewrite the integral in limit form around the singularity: \( \lim_{b\to2^-} \int_{0}^{b} \frac{1}{4-x^{2}} dx \)
03

Compute the indefinite integral

Compute the indefinite integral of the function. In this case, the antiderivative is \( \frac{1}{2} \arctan(\frac{x}{2}) \).
04

Compute the limit as b approaches 2

Evaluate the integral by computing the limit as b approaches 2 from the left. This gives \( \lim_{b\to2^-} [\frac{1}{2} \arctan(\frac{b}{2}) - \frac{1}{2} \arctan(0)] = \frac{\pi}{4} \).
05

Conclusion

Since the limit exists and is finite, the integral converges and its value is \( \frac{\pi}{4} \). It can be confirmed by the graphing utility.

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