Problem 64 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(53-66,\) make a \(u\) -substitution and integrate from \(u(a)\) to \(u(b) .\) $$\int_{\pi / 4}^{3 \pi / 4} \cot x d x$$

Step-by-Step Solution

Answer

The final result after evaluating the definite integral is \(\ln |\sin(3\pi / 4)| - \ln |\sin(\pi / 4)|\). You can further simplify by using the properties of logarithms and values of special angles.

Step 1: Identify the substitution function

We'll substitute \(u\) for \(x\), so we begin by setting \(u = x\). The differential \(du\) is simply \(dx\).

Step 2: Change the limits of integration

Since we're changing the variable we're integrating with respect to, the limits of integration also need to change. In this case, as \(u = x\), we substitute these into our limits: \(u(\pi / 4) = \pi / 4\) and \(u(3\pi / 4) = 3\pi / 4\). So our new limits of integration are \(\pi / 4\) and \(3\pi / 4\).

Step 3: Perform the u-substitution

After the substitution, the integral becomes \(\int_{\pi / 4}^{3 \pi / 4} \cot(u) du\).

Step 4: Solve the integral

The integral of \(\cot(u)\) is \(\ln |\sin(u)|\). So the result of our indefinite integral is \(\ln |\sin(u)|\), evaluated from \(\pi / 4\) to \(3\pi / 4\).

Step 5: Apply limits for the definite integral

Finally, substitute back the limits of integration back into the integral to obtain the final result: \(\ln |\sin(3\pi / 4)| - \ln |\sin(\pi / 4)|\).

Key Concepts

Definite Integral

The definite integral is a way to calculate the area under a curve between two points, known as limits of integration. By evaluating a definite integral, you can determine the total accumulation of a quantity. Unlike an indefinite integral, which represents a family of functions, a definite integral provides a specific numerical value.
In our exercise, we're evaluating the definite integral of the function \( \cot x \) from \( \pi/4 \) to \( 3\pi/4 \). By using the technique of substitution, we simplify this integration process. Once we've integrated, we apply the limits to find the total area under the curve between these two points.
By subtracting the integral value at the lower limit from the integral value at the upper limit, we finalize the computation. This is essential for determining the exact area or quantity for the specified interval.

Trigonometric Functions

Trigonometric functions, like sine, cosine, and cotangent, are fundamental in calculus, providing periodic oscillations. The specific function in our integral is \( \cot x \), which is the reciprocal of \( \tan x \). This means \( \cot x = \frac{\cos x}{\sin x} \).
Trigonometric functions have specific properties and identities that can be leveraged to simplify integration. For example, knowing that the integral of \( \cot x \) is \( \ln |\sin x| \) allows us to smoothly integrate the expression once a substitution is made. Understanding these functions helps to recognize which techniques to apply and when, leading to more efficient problem-solving.

• \( \sin x \) and \( \cos x \) are the building blocks of most trigonometric functions.
• \( \cot x \) = \( \frac{1}{\tan x} = \frac{\cos x}{\sin x} \).
• These functions are periodic, meaning they repeat values over a regular interval.

Limits of Integration

The limits of integration define the interval over which we want to find the area under a curve. In a definite integral, these limits are crucial as they specify the start and end points of our calculation.
When performing a \( u \)-substitution, it's important to adjust the limits of integration to match the new variable. In our scenario, the substitution doesn't change the original limits numerically, but in many cases, they might change and need careful computation. This ensures that calculations reflect the actual range of integration for the transformed variable.
By carefully handling these limits, you make sure the final result accurately represents the sought total of the interval in question. Always remember to revert the substitution process when computing the final numerical outcome.