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Magazine Chapter 5: Problem 26
Evaluate the integrals.$$\int_{0}^{\pi / 3}(\cos x+\sec x)^{2} d x.$$
Short Answer
Expert verified
The value of the integral is \(\frac{\sqrt{3}}{2} + \frac{\pi}{2} + \sqrt{3}\).
Step by step solution
01
Expand the Square
First, expand the integrand \((\cos x + \sec x)^2\) using the identity \((a+b)^2 = a^2 + 2ab + b^2\). Thus, we have: \[(\cos x)^2 + 2 \cos x \sec x + (\sec x)^2\].
02
Simplify the Expression
The expression \((\cos x)^2 + 2 \cos x \sec x + (\sec x)^2\) simplifies to: - Note \(\cos x \sec x = 1\), because \(\sec x = \frac{1}{\cos x}\). So, we have \((\cos x)^2 + 2 \cdot 1 + (\sec x)^2 = (\cos x)^2 + 2 + \sec^2 x\).
03
Express in Terms of Secant
Recall the Pythagorean identity: \(1 + \tan^2 x = \sec^2 x\). Thus, \((\cos x)^2 = 1 - \sin^2 x\) and \((\sec x)^2 = 1 + \tan^2 x\). So, the expression becomes \(1 - \sin^2 x + 2 + 1 + \tan^2 x = 4 - \sin^2 x + \tan^2 x\).
04
Simplify and Integrate
The integral becomes: \[\int_{0}^{\pi/3} (\cos^2 x + 2 + \sec^2 x) \, dx\]. Integrate term by term: - For \(\int \cos^2 x \, dx\), use \(\frac{1}{2}(1 + \cos 2x)\). - For \(\int \sec^2 x \, dx\), the integral is \(\tan x\).Thus, the integral becomes: \[\frac{x}{2} + \frac{1}{4}\sin 2x + 2x + \tan x\Big]_{0}^{\pi/3}\].
05
Evaluate the Definite Integral
Substitute \(x = \frac{\pi}{3}\) and \(x = 0\) in the evaluated expression.Using \(x = \pi/3\): - \( \frac{\pi}{6} + \frac{1}{4} \sin(\frac{2\pi}{3}) + \frac{2\pi}{3} + \tan(\frac{\pi}{3}) \)Using \(x = 0\): - \( 0 + 0 + 0 + 0 \) Finally, compute the expression: \( \frac{\pi}{6} + \frac{1}{4}(\frac{\sqrt{3}}{2}) + \frac{2\pi}{3} + \sqrt{3} - 0 \). Set each term's result, so the solution is the difference of these two when evaluating at the bounds.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Definite Integral
The definite integral is a key concept in calculus that represents the area under a curve for a given interval. For the integral \[ \int_{0}^{\pi/3} (\cos x + \sec x)^2 \ dx \],it signifies the area bounded by the curve \((\cos x + \sec x)^2\)between the x-values of 0 and \(\pi/3\).
To calculate a definite integral, follow these steps:
To calculate a definite integral, follow these steps:
- Find the antiderivative of the integrand function, which is a function whose derivative equals the integrand.
- Evaluate the antiderivative at the upper and lower limits of the interval.
- Subtract the value at the lower limit from the value at the upper limit.
Trigonometric Identities
Trigonometric identities are critical in simplifying integrals involving trigonometric functions. In the given integral,\((\cos x + \sec x)^2\),key identities are used to break down and simplify the expression. The specific identities used here include:
- The Pythagorean identity: \(\sec^2 x = 1 + \tan^2 x\),which helps in expressing trigonometric functions in terms of one another.
- Basic reciprocal identity: \(\sec x = \frac{1}{\cos x}\),leading to the simplification \(\cos x \sec x = 1\).
Integration Techniques
When solving integrals, various integration techniques can simplify the process, such as substitution, integration by parts, or trigonometric identities. For our exercise, the solution utilized:
- Trigonometric simplification: The integrand \((\cos x + \sec x)^2\) is expanded and then simplified using trigonometric identities.
- Term-by-term integration: Once simplified, the integral \[ \int_{0}^{\pi/3} (\cos^2 x + 2 + \sec^2 x) \ dx \]can be integrated term by term.
- The integral \(\int \cos^2 x \, dx\) is handled using the identity \(\frac{1}{2}(1 + \cos 2x)\), simplifying the process.
- The integral \(\int \sec^2 x \, dx\) results directly in \(\tan x\), which is straightforward.
