Problem 27 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-28, perform the indicated integrations. \(\int \tan ^{3} x \sec ^{2} x d x\)

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{\tan^4 x}{4} + C \).

Step 1: Identify the Integration Technique

We will use substitution to solve the integral \( \int \tan^3 x \sec^2 x \, dx \). The presence of \( \sec^2 x \, dx \) suggests that the substitution \( u = \tan x \) might be suitable.

Step 2: Perform the Substitution

Let \( u = \tan x \). Then, the derivative of \( u \) with respect to \( x \) is \( \frac{du}{dx} = \sec^2 x \), so \( du = \sec^2 x \, dx \). Substituting in the integral, we have: \[ \int u^3 \, du \].

Step 3: Integrate with Respect to the New Variable

Now, we integrate \( \int u^3 \, du \). This is a simple power rule integration: \[ \int u^3 \, du = \frac{u^4}{4} + C \], where \( C \) is the constant of integration.

Step 4: Substitute Back to the Original Variable

Replace \( u \) with \( \tan x \) to express the result in terms of \( x \): \[ \frac{u^4}{4} + C = \frac{(\tan x)^4}{4} + C \].

Step 5: Finalize the Result

The integral evaluates to \[ \int \tan^3 x \sec^2 x \, dx = \frac{\tan^4 x}{4} + C \]. Ensure that you include the constant of integration \( C \).

Key Concepts

Calculus

Calculus is a branch of mathematics that focuses on change. It provides tools to handle rates of change and accumulation.

There are two main parts: differentiation and integration. Differentiation is about finding the rate at which things change. Integration, on the other hand, deals with finding the total accumulation of quantities. Think of it as adding up small parts to get a whole.

• Differentiation tells you how things change step by step.
• Integration tells you the total effect of those changes.

Both are crucial in science and engineering because they help describe everything from motion to growth.

Here's a basic takeaway: whenever you think of handling changes, think calculus!

Indefinite Integrals

Indefinite integrals are one of the two types of integrals in calculus, the other being definite integrals.

When you compute an indefinite integral, you're essentially finding a function whose derivative gives you the original function.

• Think of it as doing the reverse of differentiation.
• It's like asking, "What function did I differentiate to end up here?"

Indefinite integrals don't have limits (beginning or end points) like definite integrals do. That's why they are called 'indefinite'.

The result of an indefinite integral includes a constant of integration, usually denoted as \( C \). This constant is important because differentiation wipes out constants. Remember, when integrating, you want to account for every possible original state of the function.

Trigonometric Functions

Trigonometric functions are fundamental in calculus. They relate angles of triangles to ratios of side lengths. These functions include sine, cosine, and tangent.

In the world of integration, they often make appearances due to their oscillatory nature.

• They can transform complex integrals into more manageable forms.
• Substitution often involves trigonometric identities to simplify integrals.

For example, in the integration problem \( \int \tan^3 x \sec^2 x \, dx \), we use the fact that the derivative of \( \tan x \) is \( \sec^2 x \). This neat relationship makes our job easier when we substitute \( u = \tan x \).

Trigonometric functions crop up in many branches of science due to their periodicity. They describe waves and oscillations perfectly. This means they are not only essential in calculus but also crucial in physics, engineering, and beyond.