Problem 4 - Thomas Calculus

Question

The integrals in Exercises \(1-40\) are in no particular order. Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate, and then use a substitution to reduce it to a standard form. $$ \int_{\pi / 4}^{\pi / 3} \frac{d x}{\cos ^{2} x \tan x} $$

Step-by-Step Solution

Answer

\(\frac{1}{2}\ln 3\)

Step 1: Simplify the Integrand

\(\frac{1}{\cos^2 x \tan x} = \frac{1}{\cos^2 x \cdot \frac{\sin x}{\cos x}} = \frac{1}{\cos x \sin x} = \frac{2}{\sin 2x} = 2\csc 2x\)

Step 2: Alternative Simplification

Actually: \(\frac{1}{\cos^2 x \tan x} = \frac{\cos x}{\cos^2 x \sin x} = \frac{1}{\cos x \sin x}\). Or write as \(\sec x \csc x\).
Using \(u = \sin x\), \(du = \cos x\,dx\) doesn't work directly. Instead, rewrite as \(\frac{\sec^2 x}{\tan x}\).

Step 3: Integrate

\(\int \frac{\sec^2 x}{\tan x}dx\). Let \(u = \tan x\), \(du = \sec^2 x\,dx\):
\(= \int \frac{du}{u} = \ln|u| + C = \ln|\tan x| + C\)

Step 4: Evaluate

\(\left[\ln|\tan x|\right]_{\pi/4}^{\pi/3} = \ln(\tan\frac{\pi}{3}) - \ln(\tan\frac{\pi}{4}) = \ln\sqrt{3} - \ln 1 = \frac{1}{2}\ln 3\)

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in calculus, especially when evaluating integrals involving trigonometric functions. These identities allow us to simplify complex expressions, making integrals easier to solve. In the original exercise, the integral included the term \(\tan x\), which can be transformed using the identity \(\tan x = \frac{\sin x}{\cos x}\).
By applying this identity, we can rewrite the integral as \[\int_{\pi / 4}^{\pi / 3} \frac{1}{\cos x \sin x} \, dx\].
This simplification effectively changes the form of the integral into one that is easier to handle.

Algebraic Methods

Algebraic methods in integral calculus involve manipulating expressions to make them more straightforward to integrate. These techniques are essential for transforming the original complex integrals into simpler forms. In the given exercise, after substituting \(\tan x = \frac{\sin x}{\cos x}\), we use algebraic manipulation to get \( \frac{1}{\cos x \sin x} \) as a simplified form of the expression.
This step is critical because it sets the stage for further simplifications or substitutions that lead to solving the integral. Such algebraic manipulation makes it easier to apply other calculus techniques like substitution or partial fraction decomposition if needed.

Integral Calculus

Integral calculus deals with the accumulation of quantities, often represented as the area under a curve. The fundamental concept is reversing differentiation, and it requires a strong understanding of functions and their derivatives.
In the exercise, the goal is to evaluate the integral over a definite interval. To do this, we apply integration techniques after simplification. These techniques can involve direct integration, substitution, or using integral tables if the function corresponds to a standard form.
Integral calculus not only provides a method for finding antiderivatives but also for computing definite integrals, which have practical applications in physics, engineering, and other fields.

Definite Integrals

Definite integrals involve evaluating the integral from one value of \(x\) to another, giving a numerical result that represents an accumulation, such as area or volume. In this exercise, the problem specifies limits of integration \(\pi/4\) to \(\pi/3\).
The solution requires using these limits after simplifying the integrand. We solve the antiderivative and then subtract the value of the integral at the lower limit from the value at the upper limit to find the area under the curve between these points.
Definite integrals play an important role in various fields by providing precise measurements of quantities and verifying the conformity of both mathematical and real-world processes.