Problem 11 - University Calculus: Early Transcendentals

Question

Evaluate the integrals. $$\int_{\pi / 4}^{3 \pi / 4} \csc \theta \cot \theta d \theta$$

Step-by-Step Solution

Answer

The integral evaluates to 0.

Step 1: Identify Integral Type

The integral \( \int \csc \theta \cot \theta \, d\theta \) is suitable for direct application of integration formulas. It is known that the derivative of \( \csc \theta \) is \( -\csc \theta \cot \theta \).

Step 2: Rewrite the Original Integral

We recognize that \( \int \csc \theta \cot \theta \, d\theta \) is the same as looking for an antiderivative of \( \csc \theta \). Thus, \( \int \csc \theta \cot \theta \, d\theta = -\csc \theta + C \), where \( C \) is the constant of integration.

Step 3: Evaluate the Definite Integral

We substitute the limits of integration into the antiderivative: \(-\csc \theta\) evaluated from \(\pi/4\) to \(3\pi/4\). Follow the calculation: \(-\csc(3\pi/4) - (-\csc(\pi/4)) = -\frac{\sqrt{2}}{1/\sqrt{2}} + \frac{\sqrt{2}}{1/\sqrt{2}} = -\sqrt{2} + \sqrt{2} = 0\).

Step 4: Simplify and Obtain the Final Result

Since \( \csc x = \frac{1}{\sin x} \), calculate the values \( \csc(\pi/4) = \frac{1}{\sin(\pi/4)} = \sqrt{2} \) and \( \csc(3\pi/4) = \frac{1}{\sin(3\pi/4)} = \sqrt{2} \). Evaluating -\csc from these points, we find \(-\csc(3\pi/4) + \csc(\pi/4) = 0 \). Therefore, the answer is 0.

Key Concepts

Definite Integral

A definite integral refers to the integral of a function over a specific interval, providing a number rather than a function. In this context, we're evaluating \[ \int_{\pi/4}^{3\pi/4} \csc \theta \cot \theta \, d \theta \]from \(\theta = \pi/4\) to \(\theta = 3\pi/4\). This process involves finding the antiderivative first and then applying the limits of integration.

• The antiderivative approach transforms a function which can then be evaluated at the boundary points.
• A substitution of these upper and lower bounds follows, simplifying the numerical value.

This technique essentially measures the total accumulation of quantities described by the function over the interval. The result here is zero, indicating that the accumulated changes cancel each other out over the interval.

Trigonometric Integrals

Trigonometric integrals involve integrating functions that are comprised of trigonometric functions such as \(\sin\), \(\cos\), \(\tan\), \(\cot\), \(\sec\), and \(\csc\). A common strategy for these integrals includes using known derivative properties of these functions.
The given exercise specifically focuses on \[ \int \csc \theta \cot \theta \, d\theta \]which employs the derivative relationship\(-\csc \theta \cot \theta = \frac{d}{d\theta}(-\csc \theta)\).

• The approach simplifies the integral by recognizing that the antiderivative of \(\csc \theta \cot \theta\) is simply \(-\csc \theta\).
• This relationship stems from understanding the derivative rules of trigonometric functions, allowing for direct integration.

It’s an efficient technique, particularly when derivatives of trigonometric functions align closely with the original integral setup.

Integration Techniques

Various techniques in integral calculus simplify the integration of complex functions. In this specific problem, recognizing standard integral format greatly reduces complexity. The use of substitution techniques and recognizing integral patterns are common methods.

• Direct Integration: Recognizing an integral that matches the derivative of another function.
• Substitution: Useful when parts of the integrand have easily discernible derivatives.
• Trigonometric Identities: Employing identities to rewrite expressions in integrable forms.

For the given problem, the integral \( \int \csc \theta \cot \theta \, d \theta \)was solved through direct application, bypassing complex manipulations. Understanding these techniques helps tackle diverse integral problems efficiently, whether it involves simple or compound trigonometric forms.