Problem 56 - Calculus of a Single Variable

Question

Evaluating a Definite Integral In Exercises \(49-56\) , evaluate the definite integral. Use a graphing utility to verify your result. $$ \int_{\pi / 8}^{\pi / 4}(\csc 2 \theta-\cot 2 \theta) d \theta $$

Step-by-Step Solution

Answer

The definite integral evaluates to -1

Step 1: Plan your approach

Start by observing that the derivative of cotangent is negative cosecant squared, which is quite close to the integrand. It suggests to apply u-substitution with u = cot(2theta).

Step 2: Perform the u-substitution

Set \( u = \cot(2\theta) \). Then \( du = -2\csc^2(2\theta) d\theta \). Multiply numerator and denominator by 2 so that the differential \( d\theta \) can be replaced in the original integral. The new limits of integration are \( \cot(\pi/4) = 1 \) (lower limit) and \( \cot(\pi/8) = \sqrt{2} + 1\) (upper limit).

Step 3: Rewrite the original integral

The original integral thus becomes: \( - \int_{1}^{\sqrt{2} + 1} (2u -1) du \).

Step 4: Perform the Integration

The integration can now be performed easily. Integrate 2u to get \( u^2 \), and 1 to get \( u \). Therefore, solution of the integral is \( - u^2 + u \).

Step 5: Evaluate Definite Integral

Substituting boundaries, \( -[(\sqrt{2} + 1)^2 - (\sqrt{2} + 1)] - [-1^2 + 1] = -1\)

Step 6: Verification with graphing utility

Plot the integrand function and the area under the curve from pi/8 to pi/4. Then compute this numerically with your graphing utility. If the result is -1, it means the manual calculation is correct

Key Concepts

U-Substitution

The process of integrating certain functions can be simplified through a technique known as u-substitution. U-substitution is akin to the reverse of the chain rule used in differentiation. It involves replacing a complex part of the function with a single variable, thereby making it easier to integrate. In the context of definite integrals, it's crucial to also convert the limits of integration by the same substitution. This ensures the integral calculations remain consistent. For the original problem, we used the substitution \( u = \cot(2\theta) \). This substitution helps simplify the integrand from trigonometric to polynomial form.

• Identify the inner function and substitute it with \( u \).
• Rewrite the differential in terms of \( du \).
• Change the integration limits inline with substitution.

Through efficient use of u-substitution, the integration process becomes much more manageable and clear.

Integration Techniques

Integration, especially with trigonometric functions, often requires specific techniques to solve effectively. The goal of these techniques is to manipulate the expression to a form that's easier to integrate. While specific functions have their own antidifferentiation formulas, transformations are often necessary. In this exercise, after substituting \( u \), the integration becomes straightforward as \( -\int (2u - 1) \, du\) transforms the previously challenging integral into a polynomial form.

• Transform complex trigonometric functions into simpler algebraic forms post substitution.
• Execute simple integration rules: \( \int u \, du = \frac{u^2}{2} + C \) and for constants \( \int du = u + C \).

Upon executing these steps, integration becomes a simpler task of applying basic formulas.

Trigonometric Functions

Trigonometric functions like sine, cosine, tangent, and their reciprocals, such as cosecant (\( \csc \theta \)) and cotangent (\( \cot \theta \)), play key roles in calculus. These functions are essential for modeling periodic phenomena. But they also can present integration challenges. In our given integral \( \int (\csc 2\theta - \cot 2\theta) \, d\theta \), understanding the relationships between functions and their derivatives was vital. For instance, the derivative of \( \cot(2\theta) \) is \(-2\csc^2(2\theta) \), revealing a strategic path for u-substitution.

• Recognize patterns and derivatives of trigonometric identities.
• Utilize these relationships to set up substitutions, simplifying integration.

Knowing these derivatives and properties is key to tackling both definite and indefinite integrals effectively.

Graphing Utility

Graphing utilities are powerful tools in both calculus learning and practice. They provide visual insights into the behavior of functions and help verify analytical results. In this exercise, the graphing utility is used to validate the manual integration's result. By plotting the function \( \csc 2\theta - \cot 2\theta \) and examining the area under the curve from \( \pi/8 \) to \( \pi/4 \), students can visually confirm the result as \(-1\).

• Graph functions to observe behavior and areas under curves.
• Use graphing utilities to calculate integrals numerically and compare against analytical solutions.

The combination of analytical and graphical methods solidifies understanding and builds confidence in solving integrals.