Problem 24 - University Calculus: Early Transcendentals

Question

Evaluate the integrals. $$\int \tan ^{2} x \sec ^{2} x d x$$

Step-by-Step Solution

Answer

\( \frac{(\tan x)^{3}}{3} + C \)

Step 1: Identify the integral form

The given integral is \( \int \tan^{2} x \sec^{2} x \, dx \). Recognize that the integrand \( \tan^{2} x \sec^{2} x \) can be rewritten using trigonometric identities.

Step 2: Use substitution

Choose a substitution that simplifies the integral. Let \( u = \tan x \), then \( \frac{du}{dx} = \sec^{2} x \), so \( du = \sec^{2} x \, dx \). This transforms the integral to \( \int u^{2} \, du \).

Step 3: Integrate using the power rule

Integrate \( \int u^{2} \, du \) using the power rule for integration, which states \( \int u^{n} \, du = \frac{u^{n+1}}{n+1} + C \). Applying this gives \( \frac{u^{3}}{3} + C \).

Step 4: Reverse substitution

Substitute back \( u = \tan x \) into the result to express the integral in terms of \( x \). This gives \( \frac{(\tan x)^{3}}{3} + C \).

Step 5: Verify the solution

Differentiate \( \frac{(\tan x)^{3}}{3} + C \) to verify it matches the original integrand. Differentiating gives \( \tan^{2} x \sec^{2} x \), confirming correctness.

Key Concepts

Trigonometric Identities

Trigonometric identities are essential in calculus, especially when working with integrals involving trigonometric functions. These identities allow us to simplify or rewrite expressions in a more manageable form.

• Basic Identities: These include fundamental relationships, such as \( \sin^2 \theta + \cos^2 \theta = 1 \). Such identities can help restructure complex expressions.
• Double Angle Identities: Useful for reducing the complexity of integrals, such as using \( \sin 2\theta = 2\sin \theta \cos \theta \).
• Squared Function Identities: Special focus on \( \tan^2x = \sec^2x - 1 \), which is pivotal when dealing with tangent and secant functions, as shown in this exercise.

In the exercise, recognizing \( \tan^2 x \) in terms of \( \sec^2x \) via \( \tan^2x = \sec^2x - 1 \) is crucial. This simplifies decision-making for substitution.

Substitution Method

The substitution method, sometimes called \( u \)-substitution, is a powerful technique for simplifying integrals by changing variables. This method is particularly helpful when dealing with composite functions.

The central idea is to represent the integrand in terms of a new variable, \( u \), which makes the integral simpler to evaluate.

• Choosing a Substitution: Look for components of the integrand that resemble derivatives, like \( \, du = \, ...dx \).
• Deriving Relationship: Establish a differential relationship. For example, if \( u = \tan x \), then \( du = \sec^2 x \, dx \).

In this exercise, letting \( u = \tan x \), turns the integral \( \int \tan^2 x \sec^2 x \, dx \) into \( \int u^2 \, du \). This transformation simplifies the problem significantly, as it allows us to apply simple algebraic integration techniques.

Power Rule for Integration

The power rule for integration is one of the foundational tools in calculus, allowing us to integrate basic polynomial functions easily. This rule states that if you want to integrate \( u^n \), where \( n \) is not equal to \(-1\), the result is \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \), where \( C \) is the integration constant.

• Simplicity: Makes it easy to find antiderivatives of polynomial terms.
• Limitations: Not directly applicable to functions with negative exponents equaling -1.
• Implementation: In the transformed integral \( \int u^2 \, du \), the rule applies straightaway, yielding \( \frac{u^3}{3} + C \).

Having executed this rule, we get \( \frac{u^3}{3} + C \). Reversing the substitution \( u = \tan x \) returns us to the variable \( x \), completing the integration task as \( \frac{(\tan x)^3}{3} + C \).

This simple yet powerful approach is a cornerstone in integral calculus.