Problem 72 - Thomas Calculus

Question

Evaluate each integral in Exercises \(71-82\) by using any technique you think is appropriate. $$ \int_{0}^{\pi / 4}(\sec x+4 \cos x)^{2} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \(4 + 6\pi\).

Step 1: Expand the Square

First, expand the square in the integrand. Using the identity \((a + b)^2 = a^2 + 2ab + b^2\), we have:\[((\sec x + 4 \cos x)^2 = \sec^2 x + 2(\sec x)(4 \cos x) + (4 \cos x)^2)\].Simplifying, we get:\[(\sec^2 x + 8 \sec x \cos x + 16 \cos^2 x)\].

Step 2: Simplify the Integrand

The integrand can be simplified further:- Recall that \(\sec x = \frac{1}{\cos x}\), thus \(\sec x \cos x = 1\).Substituting this into the expanded form, the integrand becomes:\[\sec^2 x + 8 + 16 \cos^2 x\].

Step 3: Compute the Integral

Split the integral into separate parts and integrate each one:1. \(\int_{0}^{\pi/4} \sec^2 x \, dx\) is equal to \(\tan x\) evaluated from \(0\) to \(\pi/4\), yielding \(1\).2. \(\int_{0}^{\pi/4} 8 \, dx\) results in \(8\times (\pi/4 - 0) = 2\pi\).3. \[\int_{0}^{\pi/4} 16 \cos^2 x \, dx\] requires using the identity for cosine squared:\[\cos^2 x = \frac{1 + \cos(2x)}{2}\],therefore:\[\int_{0}^{\pi/4} 16 \cos^2 x \, dx = 8\int_{0}^{\pi/4} (1 + \cos(2x)) \, dx\].

Step 4: Evaluate the Final Integral

Continue from Step 3:Expand and integrate:- \[8\int_{0}^{\pi/4} 1 \, dx = 8 \times \frac{\pi}{4} = 2\pi\].- \[8\int_{0}^{\pi/4} \cos(2x) \, dx\] requires substitution:Let \(u = 2x\), then \(du = 2dx\) or \(dx = \frac{1}{2}du\). The limits change from \(0\) to \(\pi/2\).This results in:\[4\int_{0}^{\pi/2} \cos u \, du\] which integrates to \(4\sin u\) evaluated from \(0\) to \(\pi/2\), yielding 4.Putting these together:\[2\pi + 2\pi + 4 = 4\pi + 4\].

Step 5: Combine Results

Sum all parts from previous steps:1 from \(\sec^2 x\), \(2\pi\) from 8, and \(4\pi + 4\) from the \(16 \cos^2 x\):The total result is \(1 + 2\pi + 4\pi + 4 = 4 + 6\pi\).

Key Concepts

integration techniques

Integration techniques are methods and strategies used to solve integrals, particularly those that are not straightforward or require simplification. When facing a complex integral like \( \int_{0}^{\pi / 4}(\sec x+4 \cos x)^{2} \ dx \), it's crucial to know which techniques can simplify or resolve the problem.
One basic technique is to expand and simplify expressions just like the expansion of \((a + b)^2\), which converts the problem into simpler, more manageable parts, as seen in Step 1 of the solution. Additionally, recognizing identities such as \( \sec x = \frac{1}{\cos x} \) allows simplifying the integrand further.

• Substitution: Useful when integrals include composite functions. Here, substitution is used in part of the integral to deal with trigonometric expressions efficiently.
• Splitting the Integral: The integral is divided into parts that are easier to manage. This often involves integrating constants directly and using identities for more complex functions.

By using these techniques strategically, complex integrals can be tackled in a step-by-step, organized fashion.

trigonometric integrals

Trigonometric integrals involve integrals that include trigonometric functions like sine, cosine, tangent, and secant. These integrals often require specialized techniques because of the cyclic nature of trig functions.
For example, when integrating \( \int_{0}^{\pi/4} \sec^2 x \, dx \), knowledge of the derivative of tangent, which is \( \sec^2 x \), allows for immediate evaluation.

• Trig Identities: Simplifying these integrals often involves trigonometric identities, like \( \cos^2 x = \frac{1 + \cos(2x)}{2} \), which transforms a square into more integrable parts.
• Symmetry and Periodicity: Sometimes, knowing the symmetry or periodicity of trig functions can simplify the computation or evaluation of limits.

Utilizing these properties and relationships helps break down complex expressions into manageable integrals that can be tackled with standard techniques.

definite integrals

A definite integral, such as \( \int_{0}^{\pi / 4}(\sec x+4 \cos x)^{2} \ dx \), calculates the net area under a curve over a specific interval. It differs from an indefinite integral, which finds a general form of antiderivatives.
The definite integral evaluates the result between two limits, using the Fundamental Theorem of Calculus, which connects differentiation and integration.

• Limits of Integration: These indicate the interval over which the integral is evaluated, here from \(0\) to \(\frac{\pi}{4}\).
• Application of Results: Understanding the significance of your result in context; for example, the evaluated integral gives a precise measurement (or exact area) over the defined interval.

Overall, definite integrals are powerful tools in calculus, allowing the determination of exact quantities over specific ranges, merging the abstract concept of integration with real-world applications.