Problem 25 - Calculus Early Transcendentals: Pearson New International Edition

Question

Perform the indicated integrations. $$ \int \tan ^{-3} x \sec ^{4} x d x $$

Step-by-Step Solution

Answer

The integral is \( \frac{1}{2} \tan^2 x + 2 \ln |\tan x| - \frac{1}{2} \csc^2 x + C \).

Step 1: Simplify the Integration Expression

The integral we need to solve is \( \int \tan^{-3} x \sec^4 x \, dx \). First, recall the trigonometric identities: \( \sec x = \frac{1}{\cos x} \) and \( \tan x = \frac{\sin x}{\cos x} \). Rewrite \( \tan^{-3} x \) and \( \sec^4 x \) to help in substitution. Then we have \( \tan^{-3} x = \cot^3 x = \frac{1}{\tan^3 x} \) and \( \sec^4 x = (\sec^2 x) \cdot (\sec^2 x) \).

Step 2: Use Trigonometric Identities for Substitution

Notice \( \sec^2 x - 1 = \tan^2 x \), and \( d(\tan x) = \sec^2 x \, dx \). Let \( u = \tan x \), so \( du = \sec^2 x \, dx \). This simplifies the expression as follows: \( \int \frac{\sec^4 x}{\tan^3 x} \, dx = \int \frac{(u^2 + 1)^2}{u^3} \, du \).

Step 3: Further Simplify the Expression

Simplify \( \frac{(u^2 + 1)^2}{u^3} \) which results in \( \frac{u^4 + 2u^2 + 1}{u^3} = u + \frac{2}{u} + \frac{1}{u^3} \). Therefore, the integral becomes \( \int (u + \frac{2}{u} + \frac{1}{u^3}) \, du \).

Step 4: Integrate Term by Term

Integrate each term separately: \( \int u \, du = \frac{1}{2} u^2 + C_1 \), \( \int \frac{2}{u} \, du = 2 \ln |u| + C_2 \), and \( \int \frac{1}{u^3} \, du = \int u^{-3} \, du = -\frac{1}{2} u^{-2} + C_3 \).

Step 5: Combine and Simplify the Result

Combine all parts of the integral: \( \int (u + \frac{2}{u} + \frac{1}{u^3}) \, du = \frac{1}{2} u^2 + 2 \ln |u| - \frac{1}{2} u^{-2} + C \).

Step 6: Substitute Back in Terms of x

Recall that \( u = \tan x \). Substitute back to get the integral in terms of \( x \): \( \frac{1}{2} \tan^2 x + 2 \ln |\tan x| - \frac{1}{2} \frac{1}{\tan^2 x} + C \) .

Key Concepts

Trigonometric Integrals

Trigonometric integrals often involve products of trigonometric functions. Solving these integrals requires specific techniques and understanding of trigonometric identities. Consider the integral \( \int \tan^{-3} x \sec^4 x \, dx \). At first glance, it may seem complicated, but it’s essential to recognize it as a combination of trigonometric functions.

The strategy involves expressing all trigonometric functions in terms of tangent and secant to simplify the integral.

• Using identities like \( \sec x = \frac{1}{\cos x} \) and \( \tan x = \frac{\sin x}{\cos x} \) can be helpful.
• By rewriting the integral, we can approach the solution systematically.

Mastering these concepts provides a strong foundation for tackling not just this problem but other trigonometric integrals you might encounter.

Substitution Method

The substitution method is a key technique in calculus for simplifying integrals. It is particularly useful when dealing with functions that are derivatives of one another. In our exercise, substitution plays a crucial role.

To apply substitution, we look for a suitable substitution that simplifies the integral. Here, we identify \( u = \tan x \), which simplifies the expression greatly. Using this choice:

• \( du = \sec^2 x \, dx \)
• the integral transforms into \( \int \frac{(u^2 + 1)^2}{u^3} \, du \).

This simplification is powerful as it transforms a complex trigonometric integral into a more manageable algebraic form. Through substitution, the integration process becomes more straightforward and accessible.

Trigonometric Identities

Trigonometric identities are fundamental in transforming and evaluating integrals involving trigonometric functions. These identities express one trigonometric function in terms of another, simplifying our calculations. In this exercise, a few critical identities are utilized:

• \( \sec^2 x - 1 = \tan^2 x \)
• \( d(\tan x) = \sec^2 x \, dx \)

Employing these identities helps in efficient substitution and simplification. By rewriting as \( \sec^4 x = (\sec^2 x) \cdot (\sec^2 x) \), we are able to express everything in terms of \( \tan x \), leading to an easier integration process. Understanding and using trigonometric identities are crucial skills for solving complex integrals effectively.

Indefinite Integral

An indefinite integral, unlike a definite integral, does not have limits of integration. It represents a family of functions and includes a constant of integration \( C \).

In solving an indefinite integral, as seen in the exercise:

• We integrated term by term: \( \int u \, du = \frac{1}{2} u^2 + C_1 \), \( \int \frac{2}{u} \, du = 2 \ln |u| + C_2 \), and \( \int \frac{1}{u^3} \, du = -\frac{1}{2} u^{-2} + C_3 \).

Once the integration is done, substituting back the original variable completes the solution. Remembering to add the constant \( C \) signifies the generality of the antiderivative. The result in terms of \( x \) forms the complete indefinite integral.