Problem 21 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

\( \int \tan^3 x \, dx = \frac{\tan^2 x}{2} - \log |\sec x| + C \)

Step 1: Rewrite the Integral Using Trigonometric Identity

We start by rewriting \( \tan^3 x \) as \( \tan x \cdot \tan^2 x \). Then, using the trigonometric identity \( \tan^2 x = \sec^2 x - 1 \), we can express the integral as:\[ \int \tan x (\sec^2 x - 1) \, dx\]

Step 2: Split the Integral

Next, split the integral into two separate integrals:\[ \int \tan x \sec^2 x \, dx - \int \tan x \, dx\]

Step 3: Integrate the First Part

To solve \( \int \tan x \sec^2 x \, dx \), use substitution. Let \( u = \tan x \), then \( du = \sec^2 x \, dx \).This transforms the integral into:\[ \int u \, du = \frac{u^2}{2} + C_1 = \frac{\tan^2 x}{2} + C_1\]

Step 4: Integrate the Second Part

The second integral is \( \int \tan x \, dx \), which is a standard integral:\[ \int \tan x \, dx = \log |\sec x| + C_2\]

Step 5: Combine the Results

Combine the results of the two integrals:\[ \int \tan^3 x \, dx = \frac{\tan^2 x}{2} - \log |\sec x| + C\]where \( C = C_1 - C_2 \).

Key Concepts

Trigonometric Integration

Trigonometric integration deals with integrals of trigonometric functions. These types of integrals often require specific techniques like using trigonometric identities to simplify the function. For instance, in our example, we worked with the trigonometric function \( \tan^3 x \). Instead of directly integrating, we applied a trigonometric identity to break it down into simpler parts.

Key steps include:

• Identify and apply relevant trigonometric identities. For \( \tan^3 x \), the identity \( \tan^2 x = \sec^2 x - 1 \) was useful. This helps in rewriting the integral into a form that's often easier to solve.
• Separate the integrals when possible. This can allow for a more straightforward application of other integration techniques, as seen when we split \( \int \tan x (\sec^2 x - 1) \) into two smaller integrals.

These strategies help in simplifying and solving trigonometric integrals effectively, paving the way for applying other integration techniques.

Substitution Method

The substitution method is a powerful tool in calculus used to simplify complicated integrals. It's often referred to as 'u-substitution'. The idea is to make a substitution that simplifies the integral into a basic form.

For example, when solving \( \int \tan x \sec^2 x \, dx \), we substituted \( u = \tan x \), which transformed the differential into \( du = \sec^2 x \, dx \). This changed the original, more complex integral into \( \int u \, du \), which is much simpler to evaluate.

Steps to use substitution:

• Identify a part of the integral that can be transformed using substitution.
• Define a new variable \( u \) and rewrite all parts of the integral in terms of \( u \) and \( du \).
• Perform the integration in terms of \( u \), then substitute back the original variable.

This method is especially helpful when dealing with products of functions, where differentiating one of them transforms it into a simpler form.

Calculus Integration

Calculus integration involves finding the antiderivative of a function. It's a fundamental operation in calculus, closely related to differentiation but in reverse. When you integrate a function, you often apply specific techniques suited to the forms of the functions involved.

In our example, the integration of \( \tan^3 x \) was accomplished by breaking down the expression using substitution and identities. Calculus integration is an essential skill because it allows for the calculation of areas under curves, solving differential equations, and more.

• Proper technique selection is crucial to success in integration. It often involves trial and error initially, until one becomes familiar with the range of techniques available.
• The choice between definite and indefinite integrals is context-dependent, affecting the approach and end result.

This adaptability makes integration a versatile and powerful tool in solving real-world problems.

Definite and Indefinite Integrals

Understanding the difference between definite and indefinite integrals is key in calculus. An indefinite integral represents a family of functions and includes a constant of integration \( C \), while a definite integral computes the signed area under a curve between two points.

In our example, we focused on finding an indefinite integral, \( \int \tan^3 x \ dx \), resulting in an expression plus a constant \( C \). Indefinite integrals are used to find general solutions, whereas definite integrals provide specific values for specific conditions.

• Indefinite integrals are essential when problem-solving requires a general form without specific limits.
• Definite integrals are used when you need an exact number representing accumulated change over an interval.

Both types of integrals are foundations of integral calculus, allowing for flexible solutions to a variety of mathematical problems.