Problem 71 - Thomas Calculus

Question

Evaluate each integral in Exercises \(71-82\) by using any technique you think is appropriate. $$ \int_{\pi / 4}^{3 \pi / 4}(\csc x-\cot x)^{2} d x $$

Step-by-Step Solution

Answer

The result of the integral is \(4 - \frac{\pi}{2}\).

Step 1: Expand the Expression

First, we need to expand the expression \((\csc x - \cot x)^2\). Using the binomial expansion, this becomes:\[ (\csc x - \cot x)^2 = \csc^2 x - 2\csc x \cot x + \cot^2 x \]

Step 2: Substitute Trigonometric Identities

Now, substitute the trigonometric identities into the expanded expression. Recall that:\[ \csc^2 x = 1 + \cot^2 x \]Substitute this in:\[ \csc^2 x - 2\csc x \cot x + \cot^2 x = (1 + \cot^2 x) - 2\csc x \cot x + \cot^2 x = 1 + 2\cot^2 x - 2\csc x \cot x \]

Step 3: Break Down Integral

Break the integral of the expanded expression into separate integrals:\[ \int_{\pi/4}^{3\pi/4} \left(1 + 2\cot^2 x - 2\csc x \cot x\right) \, dx = \int_{\pi/4}^{3\pi/4} 1 \, dx + 2\int_{\pi/4}^{3\pi/4} \cot^2 x \, dx - 2\int_{\pi/4}^{3\pi/4} \csc x \cot x \, dx \]

Step 4: Evaluate Each Integral Separately

Evaluate each of the integrals separately:1. \( \int_{\pi/4}^{3\pi/4} 1 \, dx = [x]_{\pi/4}^{3\pi/4} = \frac{3\pi}{4} - \frac{\pi}{4} = \frac{\pi}{2} \)2. \( 2\int_{\pi/4}^{3\pi/4} \cot^2 x \, dx = 2\left(\int \cot^2 x \, dx \right)_{\pi/4}^{3\pi/4} \) - Use the identity \(\cot^2 x = \csc^2 x - 1\) to rewrite \(\int \cot^2 x \, dx = \int (\csc^2 x - 1) \, dx\), which integrates to \(-x - \cot x\). So: - \( 2(-x - \cot x)_{\pi/4}^{3\pi/4} = 2\left(-\frac{3\pi}{4} + \frac{\pi}{4} + \cot \frac{\pi}{4} - \cot \frac{3\pi}{4}\right) \) - Since \(\cot \frac{\pi}{4} = 1\) and \(\cot \frac{3\pi}{4} = -1\), this becomes: - \( 2\left(-\frac{\pi}{2} + 2\right) = 2 \left(-\frac{\pi}{2} + 2\right) \) - Which gives \(-\pi + 4\)3. \( -2\int_{\pi/4}^{3\pi/4} \csc x \cot x \, dx = -2\left( -\csc x \right)_{\pi/4}^{3\pi/4} \) - Integration of \(\csc x \cot x\) gives \(-\csc x\): - \( 2\left( \csc \frac{\pi}{4} - \csc \frac{3\pi}{4} \right) \) - Since \(\csc \frac{\pi}{4} = \sqrt{2}\) and \(\csc \frac{3\pi}{4} = \sqrt{2}\), this evaluates to: - \( 2\left(\sqrt{2} - \sqrt{2}\right) = 0 \)

Step 5: Combine the Results

Add the results of the evaluated integrals:\[ \frac{\pi}{2} + (-\pi + 4) + 0 = \frac{\pi}{2} - \pi + 4 \]This simplifies to:\[ 4 - \frac{\pi}{2} \]

Key Concepts

Trigonometric Integrals

Trigonometric integrals are those that involve trigonometric functions such as sine, cosine, tangent, etc. When evaluating these integrals, understanding the fundamental trigonometric identities is essential. For instance, common identities like \( \csc^2 x = 1 + \cot^2 x \) can simplify complex expressions and make integrals more manageable.
Trigonometric identities can help transform integrals into a form that is easier to integrate. For example, converting \( \csc x \) and \( \cot x \) terms into identities and breaking them down into simpler components is a typical strategy. In the given problem, using the identity \( \csc^2 x = 1 + \cot^2 x \) helped in re-writing parts of the expression to facilitate easier integration.
Such techniques not only simplify the process but provide a structured method to tackle various trigonometric integrals consistently. Knowing these identities and when to apply them is crucial when working with integrals involving trigonometric functions.

Definite Integrals

Definite integrals are integrals with specific upper and lower limits. The result is a number representing the area under the curve only between those two bounds. They are very useful in calculating real-world quantities like areas, volumes, and other concepts.
The limits of integration in the problem are \( \pi/4 \) and \( 3\pi/4 \). Evaluating definite integrals often involves finding antiderivatives first, then using the Fundamental Theorem of Calculus: compute the antiderivative at the upper limit and subtract the antiderivative at the lower limit.

• The rule: \( \int_a^b f(x) \, dx = F(b) - F(a) \) where \( F(x) \) is an antiderivative of \( f(x) \).
• In the original problem, after breaking down the integral into parts, each is evaluated using these limits to arrive at a numerical answer.

Understanding how to apply limits effectively helps in obtaining accurate results and interpreting the physical meaning of the definite integral in various contexts.

Techniques of Integration

Integration techniques are methods used to find the integral of a function. Some common techniques include substitution, integration by parts, partial fractions, and trigonometric identities among others.
Choosing the right technique is often crucial. In this case, expanding trigonometric expressions and using identities is key. Transforming the expression \((\csc x - \cot x)^2\) into \(\csc^2 x - 2\csc x \cot x + \cot^2 x\) required identifying these identities and ruling out complex forms for something integrable.

• Identifying the right identities can help in simplifying and splintering the integral into manageable parts, as was demonstrated when the problem was broken down into smaller, more digestible integrals.
• Evaluating \( \int \cot^2 x \, dx \) required converting \( \cot^2 x \) into simpler expressions using \( \csc^2 x = 1 + \cot^2 x \).

Understanding when and how to apply various techniques is vital for successful integration and can drastically simplify seemingly complex problems.