Problem 45 - Calculus Early Transcendentals

Question

Determine the following indefinite integrals. Check your work by differentiation. $$\int \sec 4 \theta \tan 4 \theta d \theta$$

Step-by-Step Solution

Answer

Question: Find the indefinite integral of the function f(θ) = sec(4θ)tan(4θ). Answer: The indefinite integral of the function f(θ) = sec(4θ)tan(4θ) is ∫f(θ) dθ = (1/4)sec(4θ) + C .

Step 1: Apply the integration rule

Since we have $$\int \sec(4\theta)\tan(4\theta) d\theta$$, we can let $$u = 4\theta$$. So, $$du = 4d\theta$$. Now, divide both sides by 4 to get: $$\frac{1}{4} du = d\theta$$. Now substitute u and dθ in the integral: $$\int \sec(u)\tan(u) \cdot \frac{1}{4} du$$ Apply the integration rule: $$\frac{1}{4} \sec(u) + C$$

Step 2: Substitute back in terms of θ

Now we substitute back $$u = 4\theta$$: $$\frac{1}{4} \sec(4\theta) + C$$

Step 3: Check the answer by differentiation

Differentiate the result with respect to θ: $$\frac{d}{d\theta} \left(\frac{1}{4}\sec(4\theta) + C\right)$$ Using the chain rule with respect to θ: $$4 \cdot \frac{1}{4} \sec(4\theta) \tan(4\theta) = \sec(4\theta) \tan(4\theta)$$ The derivative matches the given function, so our solution is correct. The indefinite integral is: $$\int \sec(4\theta) \tan(4\theta) d\theta = \frac{1}{4}\sec(4\theta) + C$$

Key Concepts

Integration Techniques

Integration techniques are various methods used to find integrals of functions, especially when they can't be solved by basic formulas alone. There are several techniques such as substitution, integration by parts, partial fraction decomposition, and trigonometric integrals, each suited to different kinds of problems. For this particular problem, we're focusing on the substitution method.

Mastering different techniques is essential because they allow you to handle a wide variety of functions. Sometimes, you might need to combine multiple techniques to solve a single problem.

• **Substitution** is great for transforming the integral into a simpler form.
• **Integration by parts** is useful when dealing with products of functions.
• **Trigonometric integrals** often use identities to simplify the integral.

Getting comfortable with these techniques comes with practice, as each problem can provide a unique challenge. Always look for ways to simplify the function or relate it to a standard form.

Substitution Method

The substitution method, also known as the "u-substitution," is a powerful tool for solving integrals. It involves changing the variable of integration to simplify the integral. The main idea is to substitute part of the integrand (the expression being integrated) with a new variable, say \( u \), making the integration process easier.

In the exercise, we use substitution to tackle the integral \( \int \sec(4\theta) \tan(4\theta) d\theta \). By letting \( u = 4\theta \), the integral becomes \( \int \sec(u) \tan(u) \cdot \frac{1}{4} du \). This substitution allows us to integrate with respect to \( u \) instead of \( \theta \), simplifying our task.

• Identify a substitution that simplifies the integrand.
• Compute \( du \) and express \( d\theta \) in terms of \( du \).
• Rewrite the integral in terms of \( u \).
• Once integrated, substitute back the original variable.

The substitution method facilitates integration by transforming a complex problem into a simpler one that is more manageable. Remember to always adjust the bounds if solving a definite integral.

Differentiation

Differentiation is the process of finding the derivative of a function, which measures how the function value changes as its input changes. In the context of verifying integration results, differentiation is a crucial step that ensures accuracy.

When you integrate a function and obtain an indefinite integral, you can check your work by differentiating the result.
If you end up with the original function (or integrand), your integration is correct.

In this exercise, we differentiated \( \frac{1}{4}\sec(4\theta) + C \) with respect to \( \theta \). By applying the chain rule, the derivative \( 4 \cdot \frac{1}{4} \sec(4\theta) \tan(4\theta) \) simplifies back to the original integrand, \( \sec(4\theta) \tan(4\theta) \). This shows our integrated function is accurate.

• Ensure your differentiation follows standard rules, such as the chain rule.
• Check the derivative against the original integrand.
• Confidence in integration often comes from being able to reliably differentiate your results.

Differentiation not only verifies your work but also helps improve your overall understanding of calculus.

Trigonometric Integrals

Trigonometric integrals involve integrating functions that include trigonometric functions such as sine, cosine, tangent, and their reciprocals like secant and cosecant. These integrals can often be complicated due to the periodic and complex nature of trigonometric functions.

To solve trigonometric integrals like \( \int \sec(4\theta) \tan(4\theta) d\theta \), you often apply identities or use substitution, as seen in this exercise. Here, recognizing the derivative of \( \sec u \) as \( \sec u \tan u \) allowed us to use the substitution method effectively, leading to simpler integration.

• Utilize trigonometric identities to simplify expressions before integrating.
• Consider substitution when encountering products of trigonometric functions.
• Practice recognizing patterns in trigonometric integrals for faster solving.

Strong familiarity with trigonometric identities and differentiation of trig functions immensely aids in solving these types of integrals. It's these fundamentals that transform complex integrals into more approachable problems.