Problem 42 - Calculus Early Transcendentals

Question

Integrals involving tan \(x\) and sec \(x\) Evaluate the following integrals. $$\int \tan ^{5} \theta \sec ^{4} \theta d \theta$$

Step-by-Step Solution

Answer

Question: Evaluate the integral: $$\int \tan^5 \theta \sec^4 \theta d\theta$$ Answer: The integral evaluates to: $$\int \tan^5 \theta \sec^4 \theta d\theta = \frac{1}{10}\tan^{10}\theta + \frac{1}{4}\tan^8\theta + \frac{1}{6}\tan^6\theta + C$$

Step 1: Rewrite using the Pythagorean identity

Recall the Pythagorean identity: $$\sec^2\theta = \tan^2\theta + 1$$ Now, write the given integral in terms of \(\tan\) and \(\sec\) functions: $$\int \tan^5 \theta \sec^4 \theta d\theta$$

Step 2: Apply substitution method

Let's make a substitution: $$u = \tan\theta \Rightarrow du = \sec^2\theta d\theta$$ So our integral becomes: $$\int u^5(\sec^4\theta)d\theta$$ But we need to eliminate the secant term. Using the Pythagorean identity, we can achieve this: $$\sec^4\theta = (\sec^2 \theta)^2 = (\tan^2\theta + 1)^2$$ So, $$\int u^5(\sec^4\theta)d\theta = \int u^5(u^2 +1)^2 du$$

Step 3: Expand and integrate

Expand the expression inside the integral: $$\int u^5(u^2 +1)^2 du = \int u^5(u^4 + 2u^2 + 1) du$$ Now integrate each term separately: $$\int u^5(u^4 + 2u^2 + 1) du = \int u^9 du + 2 \int u^7 du + \int u^5 du$$ Integrate each part: $$\frac{u^{10}}{10} + 2 \frac{u^8}{8} + \frac{u^6}{6} + C = \frac{1}{10}\tan^{10}\theta + \frac{1}{4}\tan^8\theta + \frac{1}{6}\tan^6\theta + C$$ So, the final answer is: $$\int \tan^5 \theta \sec^4 \theta d\theta = \frac{1}{10}\tan^{10}\theta + \frac{1}{4}\tan^8\theta + \frac{1}{6}\tan^6\theta + C$$

Key Concepts

Pythagorean Identity

The Pythagorean Identity is one of the most fundamental identities in trigonometry, which connects the squares of sine, cosine, and secant or tangent. In its basic form, the identity is expressed as: \( \sec^2\theta = \tan^2\theta + 1 \). This identity is incredibly useful when working with integrals involving trigonometric functions, especially when you need to simplify expressions involving tangent and secant.

Using the Pythagorean Identity allows us to rewrite expressions and integrals in a more workable form. For instance, if you see an integral like \( \int \tan^5 \theta \sec^4 \theta d\theta \), you can use the identity to express \( \sec^2\theta \) in terms of \( \tan^2\theta \).

• This makes it easier to substitute variables and simplify the integral.
• The identity helps eliminate complex trigonometric expressions, smoothing the path to integration.

This identity is pivotal because it transforms equations, turning them into polynomial-like formats, which are often simpler to integrate.

Substitution Method

The Substitution Method, often referred to as "u-substitution," is a technique used in integration to simplify a complex integral into a more familiar form. It's akin to reversing the chain rule from differentiation. Here's how it works: you set a part of the integral as a new variable, \( u \), then express all other terms in terms of this new variable.

For example, in the integral \( \int \tan^5 \theta \sec^4 \theta d\theta \), you'd substitute \( u = \tan\theta \).
This substitution gives us \( du = \sec^2 \theta d\theta \), allowing us to convert the integral into terms of \( u \).

• This streamlining process can hugely simplify the given integral.
• The goal is to find an expression in which the integration becomes straightforward.

Ultimately, you integrate with respect to \( u \) and then substitute back the original variable. This method is incredibly powerful but requires a good understanding of relationships between functions and their derivatives.

Trigonometric Integrals

Trigonometric Integrals are a category of integrals that involve the trigonometric functions, such as sine, cosine, tangent, and secant. These integrals often require specific techniques to simplify and solve due to their complex periodic nature.

In solving such integrals, knowing identities and patterns is key. For example, we used a Pythagorean Identity in our solution and adoption of the substitution method simplified the integral. This is common when dealing with products of tangent and secant functions, as shown in \( \int \tan^5 \theta \sec^4 \theta d\theta \).

• They often need manipulation using identities and substitutions to be solvable.
• Patterns like those seen in polynomials often emerge, which simplifies the process further.

Trigonometric integrals may appear daunting at first, but with strategic approaches and simplifications, they can lead to neat and manageable solutions, effectively expanding one's calculus toolkit.