Problem 13 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Evaluate the integral. \(\int \tan 2 x \sec 2 x d x\)

Step-by-Step Solution

Answer

The solution to the integral is \(-\frac{1}{2} \cos^{-1} 2x + C\).

Step 1: Rewrite Integral using Trigonometric Identity

Use the identity \(\tan x = \sin x / \cos x \) to rewrite the integral. The integral now becomes \(\int \frac{\sin 2x}{\cos 2x} \cdot \frac{1}{\cos 2x} dx\), which simplifies to \(\int \frac{\sin 2x}{\cos^2 2x} dx\).

Step 2: Apply Substitution

Let \(u = \cos 2x\). Then, \(du = -2\sin 2x dx\). Thus, the given integral can be rewritten as: \(-\frac{1}{2} \int \frac{1}{u^2} du\).

Step 3: Simplify and Integrate

The expression in the integral is the derivative of \(u^{-1}\). Thus, the integral simplifies to \(-\frac{1}{2} u^{-1} + C\), where \(C\) is the constant of integration.

Step 4: Substitute Back Original Variable

Finally, substitute back the original variable \(x\), which gives the result: \(-\frac{1}{2} \cos^{-1} 2x + C\).

Key Concepts

Integration Techniques

Integration techniques are strategies used to find the integral, essentially the reverse operation of differentiation. The goal is to determine the area under the curve. Several techniques can help with different kinds of integrals, such as substitution, integration by parts, partial fraction decomposition, and trigonometric integration.

• Substitution: This is useful when an integral contains a function and its derivative. It simplifies the integration process significantly.
• Integration by Parts: Based on the product rule for differentiation and useful for products of polynomials and exponential functions.
• Trigonometric Integration: Involving integrals with trigonometric functions, using identities to simplify.
• Partial Fractions: Decomposing a rational function into simpler fractions to integrate each part.

For the integral in our exercise, the substitution method was used after rewriting the integral with trigonometric identities. It's an effective combination to tackle certain complex integrals.

Trigonometric Identities

Trigonometric identities are mathematical equations involving trigonometric functions that are universally true. These identities are vital not only for solving integrals but also for simplifying expressions. Learning and knowing these identities makes integration easier.
In the given exercise, the identity \(\tan x = \frac{\sin x}{\cos x} \)is used to simplify the integral. Transforming the integral \(\int \tan 2x \sec 2x \, dx\) into \(\int \frac{\sin 2x}{\cos^2 2x} \, dx\) was a crucial step.
Recognizing these identities helps in breaking down complex trigonometric integrals into manageable pieces. Mastering these will empower you to breeze through many integration challenges.

Substitution Method

The substitution method, or "u-substitution," is a technique that simplifies finding integrals, especially when integrals contain compositions of functions. This method is akin to reverse chain rule used in differentiation. By substituting part of the integral with a single variable, it transforms the integral into a simpler form.During substitution, you select a substitution variable, usually noted as \(u\). You will also need to change \(dx\) to \(du\) by differentiating \(u\) with respect to \(x\).
For the exercise, they let \(u = \cos 2x\) and found \(du = -2\sin 2x \, dx\). Rewriting the integral in terms of \(u\) gives \(-\frac{1}{2} \int \frac{1}{u^2} \, du\). This change made the integral straightforward to solve. The substitution method is powerful because it often turns a seemingly tough integral into a manageable one.

Definite Integrals

A definite integral calculates the net area under the curve between two points on the x-axis. It has limits of integration, typically denoted as \(a\) and \(b\). Unlike indefinite integrals, the result of a definite integral is a number representing this area.The fundamental theorem of calculus bridges the gap between differentiation and integration. It shows that integration is the inverse operation of differentiation:

• Part 1: If \(F\) is an antiderivative of \(f\) on an interval \([a, b]\), then \(\int_a^b f(x) \, dx = F(b) - F(a)\).
• Part 2: If \(f\) is continuous on \([a, b]\), then it has an antiderivative \(F\) on that interval, and its definite integral calculates changes in \(F\) over the interval \([a, b]\).

This integral involved was indefinite, as no limits were provided. However, understanding definite integrals is essential, as they are prevalent in various applications.