Problem 21 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-28, perform the indicated integrations. \(\int \tan ^{3} x d x\)

Step-by-Step Solution

Answer

The integral of \( \tan^3 x \) is \( \frac{\tan^2 x}{2} - \ln |\cos x| + C \).

Step 1: Rewrite the Integral

The first step in solving the integral \( \int \tan^3 x \, dx \) is to rewrite \( \tan^3 x \) in terms of \( \tan x \) and \( \sec^2 x \). Notice that \( \tan^3 x = \tan x \cdot \tan^2 x \). We can also use the identity \( \tan^2 x = \sec^2 x - 1 \). Hence, rewrite the integrand as \( \tan x (\sec^2 x - 1) \).

Step 2: Setup the Integral with the New Expression

Substitute the identity into the integral to get:\[ \int \tan^3 x \, dx = \int \tan x (\sec^2 x - 1) \, dx = \int (\tan x \sec^2 x - \tan x) \, dx. \]

Step 3: Split the Integral

Now split the integral into two separate integrals:\[ \int (\tan x \sec^2 x - \tan x) \, dx = \int \tan x \sec^2 x \, dx - \int \tan x \, dx. \]

Step 4: Integrate the First Term

Focus on the first integral \( \int \tan x \sec^2 x \, dx \). Notice that the derivative of \( \tan x \) is \( \sec^2 x \). So, use substitution: let \( u = \tan x \), which gives \( du = \sec^2 x \, dx \).The integral becomes:\[ \int u \, du = \frac{u^2}{2} = \frac{\tan^2 x}{2}. \]

Step 5: Integrate the Second Term

The second integral is \( \int \tan x \, dx \). We know that \( \tan x = \frac{\sin x}{\cos x} \), and use substitution: let \( u = \cos x \), which gives \( du = -\sin x \, dx \). Thus, the integral becomes:\[ -\int \frac{1}{u} \, du = -\ln |u| + C = -\ln |\cos x| + C. \]

Step 6: Combine the Results

Combine the results from the two integrals:\[ \int \tan^3 x \, dx = \frac{\tan^2 x}{2} - \ln |\cos x| + C. \]

Step 7: Write the Final Answer

With all the components combined, the final solution for \( \int \tan^3 x \, dx \) is:\[ \int \tan^3 x \, dx = \frac{\tan^2 x}{2} - \ln |\cos x| + C. \]

Key Concepts

Integration Techniques

Integration techniques play a vital role in calculus, especially when dealing with complex expressions. The art of integration involves finding a function whose derivative is the given function. This process is the reverse of differentiation. Understanding various integration techniques helps tackle different kinds of integrals.

When solving integrals, you may encounter expressions that are not straightforward to integrate. In such scenarios, advanced techniques, like substitution or integration by parts, become valuable tools. For instance, in our problem involving \( \int \tan^3 x \, dx \), rewriting the expression to remove complexities allows for simpler integration steps. Identifying identities, splitting integrals, and substituting variables are some handy approaches to help streamline the process and achieve accurate results.

Trigonometric Integrals

Trigonometric integrals involve functions of sine, cosine, tangent, and their hyperbolic counterparts. These integrals require special attention due to the cyclic nature and complex interrelationships among trigonometric functions.

In the exercise of integrating \( \int \tan^3 x \, dx \), understanding trigonometric identities is crucial. The identity \( \tan^2 x = \sec^2 x - 1 \) allows us to express the integral in a form that's easier to handle. Breaking down compound trigonometric expressions often simplifies the integration process. Recognizing these relationships is key in evaluating trigonometric integrals effectively.

It's practical to become familiar with common trigonometric identities. Doing so equips you to transform and simplify integrals, making them more approachable.

Substitution Method

The substitution method, also called "u-substitution," is a powerful technique for solving integrals. It involves changing variables to simplify an integral's evaluation. This approach takes advantage of the chain rule from differentiation inversely.

In the example of \( \int \tan x \sec^2 x \, dx \), let \( u = \tan x \), making \( du = \sec^2 x \, dx \). When the integral is reformed into \( \int u \, du \), it's a straightforward power integral \( \frac{u^2}{2} \), or in terms of \( x \), \( \frac{\tan^2 x}{2} \).

This method greatly reduces complexity by transforming the integral into a more recognizable form. With the correct substitution, the integration becomes considerably more straightforward. Mastery of the substitution method is crucial for calculus students, as it is widely applicable across a variety of integration problems.

Integral Identities

Integral identities are crucial tools in simplifying and solving calculus problems. They include relationships based on the properties of functions, often simplifying trigonometric expressions into integral-friendly forms.

In integrating \( \int \tan^3 x \, dx \), the identity \( \tan^2 x = \sec^2 x - 1 \) is instrumental. By substituting this identity into the integral, the problem becomes approachable by breaking it down into more straightforward parts. Simplifying complex expressions into basic identities helps exploit the fundamental properties of calculus, leading to correct and simplified solutions.

Familiarity with common integral identities and their applications not only saves time but also assists in visualizing and solving complex integrals. These can provide new perspectives and tools for tackling integrations that seem daunting at first glance.