Problem 39 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

Question: Evaluate the definite integral: $$\int_{0}^{\pi / 2} \sec x \tan x d x$$ Answer: The integral diverges. In other words, it does not have a finite value.

Step 1: Choose the substitution

We will let \(u = \sec x\). This simplifies our integral and provides an easy differentiation for the dx substitution.

Step 2: Differentiate the substitution

Differentiate \(u\) with respect to \(x\): $$\frac{d u}{d x} = \frac{d(\sec x)}{d x} = \sec x \tan x$$

Step 3: Solve for dx

Now we can solve for \(dx\): $$d x = \frac{d u}{\sec x \tan x}$$

Step 4: Replace variables and dx in the integral

We can now replace \(\sec x \tan x\) and \(dx\) in the original integral: $$\int_{0}^{\pi / 2} \sec x \tan x d x = \int \frac{1}{\sec x \tan x} d(\sec x)$$ Using the substitution \(u = \sec x\), this becomes: $$\int \frac{1}{u} d u$$

Step 5: Evaluate the integral

This integral can now be easily evaluated: $$\int \frac{1}{u} d u = \ln |u| + C$$

Step 6: Replace u with the original variable

Now replace \(u\) with the original variable \(\sec x\): $$\ln |\sec x| + C$$

Step 7: Compute within the limits

Finally, compute the definite integral within the given limits \(0\) and \(\pi/2\): $$\left[\ln |\sec x|\right]^{\pi/2}_0 = \left(\ln |\sec(\pi/2)|\right) - \left(\ln |\sec(0)|\right)$$ This simplifies to: $$\ln \left| \frac{\sec(\pi/2)}{\sec(0)} \right| = \ln \left| \frac{\infty}{1} \right| = \infty$$ Since the result is \(\infty\), the integral is said to diverge.