Textbook Question
Use symmetry to explain why.
∫⁴₋₄ (5𝓍⁴ + 3𝓍³ + 2𝓍² + 𝓍 + 1) d𝓍 = 2 ∫₀⁴ (5𝓍⁴ + 2𝓍² + 𝓍 + 1) d𝓍 .
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8. Definite Integrals
Introduction to Definite Integrals
3:07 minutesProblem 5.4.21
Textbook Question
Symmetry in integrals Use symmetry to evaluate the following integrals.
∫₋π/₄^π/⁴ sec² x dx
Verified step by step guidance1
Recognize that the integral ∫₋π/₄^π/⁴ sec² x dx involves a symmetric interval about the origin, specifically from -π/4 to π/4. This suggests that symmetry properties of the function sec²(x) can be used to simplify the evaluation.
Determine whether the function sec²(x) is even or odd. Recall that a function f(x) is even if f(-x) = f(x) and odd if f(-x) = -f(x). For sec²(x), since sec(-x) = sec(x), it follows that sec²(-x) = sec²(x), making sec²(x) an even function.
Use the property of even functions in integrals: If f(x) is even, then ∫₋a^a f(x) dx = 2∫₀^a f(x) dx. Apply this property to the given integral, rewriting it as 2∫₀^π/₄ sec²(x) dx.
Recall the antiderivative of sec²(x), which is tan(x). Use this to express the integral ∫₀^π/₄ sec²(x) dx as tan(x) evaluated from 0 to π/4.
Substitute the limits of integration into tan(x): tan(π/4) and tan(0). Simplify the expression to complete the evaluation of the integral.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Symmetry in Integrals
Symmetry in integrals refers to the property that allows certain integrals to be simplified based on the symmetry of the function being integrated. If a function is even, meaning f(-x) = f(x), the integral from -a to a can be simplified to 2 times the integral from 0 to a. Conversely, if a function is odd, where f(-x) = -f(x), the integral over a symmetric interval around zero equals zero.
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Secant Function
The secant function, denoted as sec(x), is the reciprocal of the cosine function, expressed as sec(x) = 1/cos(x). It is important in calculus, particularly in integration and differentiation, as it appears frequently in trigonometric integrals. Understanding the behavior of sec(x) and its derivatives is crucial for evaluating integrals involving this function.
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Definite Integrals
A definite integral represents the signed area under a curve defined by a function over a specific interval [a, b]. It is calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. Evaluating definite integrals often involves finding antiderivatives and applying limits, making it essential to understand this process for solving integral problems.
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