Problem 33
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. $$ \int_{0}^{\pi / 2} \tan \theta d \theta $$
Step-by-Step Solution
Verified Answer
The improper integral diverges.
1Step 1: Understand the function behavior
As \(\theta\) approaches \(\pi / 2\) from the left, \(\tan \theta\) approaches \(\infty\). Thus, the integral has an infinite discontinuity at \(\pi / 2\), and hence it is an improper integral.
2Step 2: Solve the Integral
We use the result that the integral of \(\tan \theta\) is \(-\ln |\cos \theta|\), so we have \(-\ln |\cos \theta|\) for \(0\) to \(\pi / 2\). Evaluation at the limits gives \(-\ln |\cos (\pi / 2)| + \ln |\cos (0)| = -\ln |0| + \ln |1| = -\ln 0\). Since \(\ln 0\) is undefined, the integral is also undefined.
3Step 3: Convergence/Divergence
Because the integral is undefined, we say that the improper integral diverges.
Other exercises in this chapter
Problem 32
Evaluate the definite integral. Use a graphing utility to confirm your result. $$ \int_{0}^{\pi} x \sin 2 x d x $$
View solution Problem 33
Solve the differential equation. \(\frac{d r}{d \theta}=\sin ^{4} \pi \theta\)
View solution Problem 33
In Exercises 33-36, find or evaluate the integral. $$ \int_{0}^{\pi / 2} \frac{1}{1+\sin \theta+\cos \theta} d \theta $$
View solution Problem 33
In Exercises \(27-38,\) (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L'Hôpital's
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