Question

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

Short answer

Answer: The integral diverges.

Step-by-step solution

Finding the antiderivative of the secant function

Recall that the antiderivative of the secant function is: $$\int \sec x dx = \ln | \sec x + \tan x | + C$$

Applying the limits of integration

Now we will apply the limits of integration to the antiderivative we found in the previous step: $$\int_{0}^{\pi/2} \sec \theta d\theta = \left[ \ln |\sec \theta + \tan \theta| \right]_{0}^{\pi/2}$$

Evaluating the antiderivative at the limits of integration

Evaluate the antiderivative function at the upper limit and then at the lower limit: Upper limit evaluation: $$\ln |\sec(\pi/2) + \tan(\pi/2)|$$ Since \(\sec(\pi/2) = \frac{1}{\cos(\pi/2)} = \frac{1}{0}\), it is undefined and results in a infinite value. So, evaluating the function at the upper limit is impossible. Lower limit evaluation: $$\ln | \sec(0) + \tan(0) | = \ln |1+0|= \ln 1 = 0$$ However, evaluating at the lower limit doesn't help us since we could not evaluate at the upper limit.

Determine if the integral converges or diverges

Since we found that the evaluation at the upper limit of integration is infinite, it follows that the integral diverges. The answer is that the integral diverges.

Key concepts

Improper Integrals

Improper integrals are special kinds of integrals where either the limits of integration are infinite, or the integrand becomes unbounded within the integration limits.Understanding improper integrals is crucial, as they help you calculate areas and values that aren't finite in the traditional sense.The integral you are asked to evaluate is an improper integral because the value of the integrand, \( \sec \theta \), becomes infinitely large as \( \theta \) approaches \( \pi/2 \).To determine whether an improper integral converges or diverges, you follow the limits approach:

• If the limit exists and is a real number, the integral converges.
• If the limit approaches infinity or does not exist, the integral diverges.

This is why evaluating at \( \theta = \pi/2 \), yields an undefined result indicative of divergence.

Antiderivative of Secant Function

Finding the antiderivative of the secant function is a must for solving the given integral.The general form of the antiderivative of the secant function is:\[\int \sec x \, dx = \ln | \sec x + \tan x | + C\]A great way to understand this antiderivative is to remember that it emerges because the derivative of \( \ln | \sec x + \tan x | \) is precisely \( \sec x \).This relationship helps us to set up steps for evaluating definite integrals involving \( \sec \theta \).The solution involves plugging this antiderivative back into the original integral with the limits of integration in place.

Limits of Integration

When dealing with definite integrals, limits of integration play a critical role. They determine the interval over which the function is integrated.In the problem at hand, the limits are from \(0\) to \(\pi/2\).Evaluating such an integral involves:

• Substituting the upper limit into the antiderivative.
• Subtracting the substituted lower limit from the result of the upper limit.

When applying limits of integration to an antiderivative where undefined values occur, like at \( \theta = \pi/2 \), you encounter improper integral behavior.This results in an infinitely large value, indicating the integral diverges.Such situations illustrate why understanding and computing limits is essential in calculus.