Problem 34 - Essential Calculus

Question

Find or evaluate the integral. $$ \int_{0}^{\pi / 2} \frac{1}{3-2 \cos \theta} d \theta $$

Step-by-Step Solution

Answer

The value of the integral is \( \frac{1}{3} \ln(\frac{3}{2}) \)

Step 1: Substitution

Substitute \( u = 2 - 3 \cos \theta \), then differentiate both sides to find \( du = 3 \sin{\theta} d\theta \)

Step 2: Transform limits of integration

Transform the original limits of integration in terms of u, when \( \theta = 0 \), \( u = 2 \) and when \( \theta = \frac{\pi}{2} \), \( u = 3 \)

Step 3: Rescale integral

Rewrite the original integral in terms of u and adjust the integral to get \( \frac{1}{3} \int_{2}^{3} \frac{1}{u} du \)

Step 4: Integration

Evaluate the integral to get \( \frac{1}{3}( \ln |3| - \ln |2| ) \)

Step 5: Simplifying

Use the properties of logarithms to simplify the result to \( \frac{1}{3} \ln(\frac{3}{2}) \)