Problem 36 - Calculus Early Transcendentals

Question

Integrals involving tan \(x\) and sec \(x\) Evaluate the following integrals. $$\int \frac{\sec ^{2} x}{\tan ^{5} x} d x$$

Step-by-Step Solution

Answer

Question: Evaluate the given integral: $$\int \frac{\sec ^{2} x}{\tan ^{5} x} d x$$ Answer: The integral can be evaluated as follows: $$\int \frac{\sec ^{2} x}{\tan ^{5} x} d x = -\frac{1}{4 \tan^4 x} + C$$

Step 1: Perform u-substitution

Choose \(u = \tan x\). Now find the derivative of \(u\) with respect to \(x\): $$\frac{du}{dx} = \frac{d(\tan x)}{dx} = \sec^2 x$$ Now, express \(dx\) in terms of \(du\): $$dx = \frac{du}{\sec^2 x}$$

Step 2: Substitute u and dx into the integral

Replace \(\tan x\) with \(u\) and also replace \(dx\): $$\int \frac{\sec ^{2} x}{\tan ^{5} x} d x = \int \frac{\sec^2 x}{u^5} \frac{du}{\sec^2 x}$$ Now, simplify the integral: $$\int \frac{1}{u^5} du$$

Step 3: Integrate with respect to u

Now, integrate the function concerning \(u\): $$\int \frac{1}{u^5} du = -\frac{1}{4u^4} + C$$

Step 4: Revert back to x

Express the result in terms of \(x\) by substituting \(u = \tan x\): $$-\frac{1}{4u^4} + C = -\frac{1}{4 \tan^4 x} + C$$ Thus, the integral is: $$\int \frac{\sec ^{2} x}{\tan ^{5} x} d x = -\frac{1}{4 \tan^4 x} + C$$

Key Concepts

U-Substitution

When tackling integrals, especially those involving trigonometric functions, one powerful method to simplify the process is u-substitution. This technique involves replacing a part of the integral with a new variable, typically u, which then simplifies the integral into a form that's easier to manage.

Using u-substitution, we first identify a function within the integral that we can differentiate to find a derivative that's also present in the integral. In the given problem, \( u = \tan x \) was chosen because its derivative, \( \sec^2 x \), was present in the integrand. This foresight reduces the integral into a simpler expression in terms of u, making it easier to integrate. This step not only streamlines the calculation but is also an essential part of the integration process when dealing with more complex functions.

Integration Techniques

There are several integration techniques that students can use to solve integrals, such as the Power Rule, Integration by Parts, Partial Fractions, and Trigonometric Integration. The latter is particularly relevant when dealing with trigonometric functions like tangent and secant.

Why Choose U-Substitution?

In our problem, u-substitution was chosen as the integration technique to handle the trigonometric integral. This is because the integral involved a complex expression of trigonometric functions that became significantly simpler after applying u-substitution. Recognizing when and how to apply these techniques is crucial for solving integrals effectively and is often dictated by the form of the function being integrated.

Trigonometric Integrals

Integrals that involve trigonometric functions, like sine, cosine, tangent, or secant, are known as trigonometric integrals. These integrals often require specific strategies to solve, such as trigonometric identities, u-substitution, or even rewriting the integral using trigonometric properties to simplify it.

Complexity of Trigonometric Functions

Some trigonometric integrals become complex due to high powers or products of different trigonometric functions. That's where certain rewrite strategies using identities like \( \tan^2 x = \sec^2 x - 1 \) or \( \sin^2 x + \cos^2 x = 1 \) can come in handy to simplify the integral before applying techniques like u-substitution.

Indefinite Integrals

An indefinite integral is a function that gives all the antiderivatives of the original function being integrated. It includes a constant term C since the antiderivative is not unique due to the derivative of a constant being zero. In our exercise, after the u-substitution and integration were performed, the result was an indefinite integral.

Expression of C

It's important to remember to add the constant term C at the end of the integration process, which represents all possible vertical shifts of the antiderivative function. This constant is vital since it encompasses all potential solutions to the integral, and its determination usually requires additional conditions such as initial values or boundary conditions in more complex scenarios.