Problem 48 - Essential Calculus

Question

Find the integral. Use a computer algebra system to confirm your result. $$ \int \frac{\sin ^{2} x-\cos ^{2} x}{\cos x} d x $$

Step-by-Step Solution

Answer

The integral of \( \frac{\sin ^{2} x-\cos ^{2} x}{\cos x} dx \) is \( \ln|\sec x + \tan x| - 2\sin x + C \)

Step 1: Rewrite using Trigonometric Identity

The first step in solving this integral is to rewrite it using trigonometric identities. Because \( \sin^2x + \cos^2x = 1 \), we can express \( \sin^2x \) as \( 1 - \cos^2x \). So the integral becomes: \( \int \frac{(1 - 2\cos^2x)}{\cos x} dx \)

Step 2: Separate the fraction

Next, separate the fraction into two separate fractions that can be integrated directly: \( \int \frac{1}{\cos x} dx - \int \frac{2\cos^2x}{\cos x} dx = \int \sec x dx - 2\int \cos x dx \)

Step 3: Compute each integral

Now compute each of these integrals separately. The integral of secant x is ln|sec x + tan x| and the integral of cos x is sin x. This gives: \( \ln|\sec x + \tan x| - 2\sin x + C \)

Key Concepts

Trigonometric Identities

Trigonometric identities are crucial tools in solving integral calculus problems, especially those involving trigonometric functions. One of the most famous trigonometric identities is \( \sin^2 x + \cos^2 x = 1 \). This identity helps simplify expressions by expressing one trigonometric function in terms of another.

In our original exercise, we used this identity to rewrite \( \sin^2 x \) as \( 1 - \cos^2 x \). This transformation makes the integral easier to handle by reducing the complexity of the expression. Understanding these identities allows us to manipulate functions into more tractable forms.

Other useful trigonometric identities include double angle formulas, such as \( \cos 2x = 1 - 2\sin^2 x = 2\cos^2 x - 1 \), which can similarly help simplify integrals. Mastery of these identities is crucial when dealing with calculus problems, as they often allow us to transform challenging integrals into more straightforward ones.

Computer Algebra Systems

Computer algebra systems (CAS), like Mathematica and Wolfram Alpha, are amazing tools for confirming algebraic manipulations and integral calculus computations. These systems can handle both simple and complex calculations, offering step-by-step solutions to integrals, derivatives, and more.

In the context of the original problem, a CAS can verify the manual integration steps by processing the rewritten integral expression and calculating its antiderivative. This verification process ensures accuracy in more intricate problems, where human error might occur.

Benefits of using a CAS include:

• Automation of algebraic manipulations, saving time and effort.
• Checking solutions against computed results for consistency.
• Exploring alternative methods and solutions for deeper understanding.

While using a CAS can provide rapid solutions, it's still essential to grasp the fundamental concepts and manual techniques of calculus. This foundational knowledge helps interpret the results and understand how the CAS arrived at a particular solution.

Indefinite Integrals

Indefinite integrals represent a fundamental concept in integral calculus, differing from definite integrals by focusing on finding an antiderivative without specific limits. When computing an indefinite integral, you determine a general formula that represents a family of functions, including an arbitrary constant, \( C \).

In the exercise, we solved for the indefinite integral \( \int \sec x dx - 2\int \cos x dx \). Each part of the expression is integrated separately, yielding the antiderivative \( \ln|\sec x + \tan x| \) and \( -2\sin x \) respectively, plus a constant \( C \).

Understanding indefinite integrals involves:

• Recognizing standard forms and patterns.
• Applying integration rules, such as substitution or integration by parts.
• Adding the constant of integration \( C \), acknowledging the family of solutions.

Indefinite integrals are essential in various applications, from calculating areas under curves to solving differential equations, making them a foundational skill in mathematics and beyond.