/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 36 Evaluate the following integrals... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 36

Evaluate the following integrals or state that they diverge. $$\int_{0}^{\pi / 2} \tan \theta d \theta$$

Short Answer

Expert verified
Answer: The definite integral \(\int_{0}^{\pi / 2} \tan \theta d \theta\) diverges.

Step by step solution

01

Finding the antiderivative of \(\tan \theta\)

To find the antiderivative of \(\tan \theta\), we can rewrite it as \(\frac{\sin \theta}{\cos \theta}\) and then use substitution method. Let \(u = \cos \theta\) and \(du = -\sin \theta d\theta\). Therefore, the antiderivative becomes: $$\int \frac{\sin \theta}{\cos \theta} d\theta = -\int \frac{1}{u} du$$ Now we can easily integrate: $$\int \frac{1}{u} du = \ln|u|+C = -\ln|\cos \theta|+C$$
02

Evaluating the definite integral using the Fundamental Theorem of Calculus

Now we need to apply the Fundamental Theorem of Calculus to evaluate the definite integral. The theorem states that if a function \(f\) is continuous on the interval \([a, b]\) and \(F\) is an antiderivative of \(f\) on that interval, then: $$\int_{a}^{b} f(x) dx = F(b) - F(a)$$ In this case, we have \(a = 0\), \(b = \frac{\pi}{2}\), and \(F(\theta) = -\ln|\cos \theta|\). Therefore, the definite integral becomes: $$\int_{0}^{\pi / 2} \tan \theta d \theta = -\ln|\cos (\pi/2)| + \ln|\cos 0|$$
03

Evaluating the range of the function

Now we need to evaluate the range of the function \(\cos \theta\) on the interval \([0, \frac{\pi}{2}]\). On this interval, the function \(\cos \theta\) ranges between \(0\) and \(1\). Therefore, when plugging in the endpoints of the interval \(a = 0\) and \(b = \frac{\pi}{2}\) we have: $$-\ln|\cos (\pi/2)| + \ln|\cos 0| = -\ln|0| + \ln|1|$$ Since the natural logarithm of \(0\) is undefined, the integral diverges. The definite integral diverges: $$\int_{0}^{\pi / 2} \tan \theta d \theta = \text{Divergent}$$

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