Problem 3 - Thomas Calculus

Question

The integrals in Exercises \(1-44\) are in no particular order. Evaluate each integral using any algebraic method or trigonometric identity you think is appropriate. When necessary, use a substitution to reduce it to a standard form. $$ \int(\sec x-\tan x)^{2} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \(-x + 2\tan x - 2\sec x + C\).

Step 1: Expand the Expression

First, expand the integrand \((\sec x - \tan x)^{2}\). Applying the formula \((a - b)^{2} = a^{2} - 2ab + b^{2}\), we get: \(\sec^{2}x - 2\sec x \tan x + \tan^{2}x\).

Step 2: Simplify Using Identity

Recall the trigonometric identities \(\sec^{2}x = 1 + \tan^{2}x\). Substitute \(\sec^{2}x\) with \(1 + \tan^{2}x\) into the expanded expression: \((1 + \tan^{2}x) - 2\sec x \tan x + \tan^{2}x = 1 + 2\tan^{2}x - 2\sec x \tan x\).

Step 3: Integrate Term by Term

Now integrate each term separately: \(\int (1 + 2\tan^{2}x - 2\sec x \tan x) \, dx\). This becomes: \(\int 1 \, dx + 2\int \tan^{2}x \, dx - 2\int \sec x \tan x \, dx\).

Step 4: Evaluate Each Integral

Evaluate \(\int 1 \, dx = x\). For \(\int \tan^{2}x \, dx\), use the identity \(\tan^{2}x = \sec^{2}x - 1\): \(\int (\sec^{2}x - 1) \, dx = \int \sec^{2}x \, dx - \int 1 \, dx\). This yields: \(\tan x - x\). \(\int \sec x \tan x \, dx\) is directly evaluated as \(\sec x\).

Step 5: Combine Results

The expression after integration is: \(x + 2(\tan x - x) - 2\sec x + C\), where \(C\) is the constant of integration. Simplifying, \(-x + 2\tan x - 2\sec x + C\) is obtained.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that are always true for any angle. They help simplify complex expressions and play a crucial role in integration processes, especially when dealing with trigonometric functions. In our problem, the identity \( \sec^2x = 1 + \tan^2x \) is especially useful. This identity allows us to rewrite complex trigonometric expressions in simpler forms, which are easier to integrate. Here, it simplifies the expanded form of \( (\sec x - \tan x)^2 \) by expressing \( \sec^2x \) in terms of \( \tan^2x \).
By applying such identities, you turn intricate functions into manageable pieces, making problem-solving much more approachable. These identities also serve to double-check your work because they are universally valid.

Substitution Method

The substitution method in integration is a powerful technique used to simplify an integral by replacing part of the integrand with a new variable. This technique is especially helpful when dealing with integrals that contain complex functions. It transforms an unmanageable integral into a simpler, standard form. Substitution acts like reverse chain rule processing by essentially undoing the derivative back into its antiderivative form. In the integration of trigonometric expressions, one might let \( u = \tan x \) or \( u = \sec x \), depending on the structure of the integrand.
This variable change makes integration easier by using simpler integral forms. When properly applied, substitution can turn a challenging problem into a straightforward one, making it an indispensable tool in your calculus toolkit.

Standard Integral Forms

Standard integral forms are predefined integrals of basic functions that you often encounter in calculus. These forms are like templates, helping you quickly calculate integrals of expressions that match their structure. Recognizing these patterns speeds up the process of finding antiderivatives. For the problem at hand, we used the standard integral form of \( \int \sec^2x \, dx = \tan x + C \), which is derived from the derivative of the tangent function. Another form involved is \( \int \sec x \tan x \, dx = \sec x + C \), linked to the derivative of the secant function.
Memorizing these common forms allows you to effortlessly solve integrals by simply identifying and matching them to your problem, thus saving you time and avoiding unnecessary complexity during calculations.

Trigonometry Integration

Trigonometry integration involves the use of trigonometric identities and standard integral forms to evaluate integrals that involve trigonometric functions. This specific approach to integration requires a good understanding of trigonometric functions and their properties. Our problem initially expanded the expression \( (\sec x - \tan x)^2 \) using the formula \((a - b)^2 = a^2 - 2ab + b^2 \). This step turned a potentially confusing function into a sum of more recognizable terms.
The next step was breaking down the integral into simpler parts using trigonometric identities. Each part was then evaluated using standard integral forms, effectively solving the integral piece by piece. This shows the power of combining identities, substitution, and standard forms to tackle problems that seem challenging at first.