Problem 40 - Essential Calculus

Question

Evaluate the definite integral. Use a graphing utility to verify your result. $$ \int_{0.1}^{0.2}(\csc 2 \theta-\cot 2 \theta)^{2} d \theta $$

Step-by-Step Solution

Answer

The result of the definite integral \(\int_{0.1}^{0.2}(\csc 2 \theta-\cot 2 \theta)^{2} d \theta \) is 0.1

Step 1: Simplify the integral

We can rewrite the function \( (\csc 2 \theta-\cot 2 \theta)^{2} \) as a sum of squares because \( \csc^{2} \theta = 1 + \cot^{2} \theta \). So the integral becomes: \( \int_{0.1}^{0.2} (1 + 2 \cot^{2} 2\theta - 2 \csc 2\theta \cot 2\theta) d \theta \)

Step 2: Split the Integral

Split the integral into three parts according to the terms in the brackets: \( \int_{0.1}^{0.2} 1d\theta + \int_{0.1}^{0.2} 2 \cot^{2} 2\theta d\theta - \int_{0.1}^{0.2} 2\csc 2\theta \cot 2\theta d\theta \)

Step 3: Evaluate each integral

The first integral can be done directly to give \( \theta \Big|_{0.1}^{0.2} \) which evaluates to 0.1 . The second integral converted into \(\frac{2}{\sin^2 2\theta} d\theta \) will require use of the power-reduction identity. The third integral is converted into \(-\frac{2}{\sin^2 2\theta} d\theta \) and the integrals cancel out.

Step 4: Summarize

Adding up all the solutions, the final result is 0.1