Textbook Question
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
β«Ο/β^Ο/Β² (cos π) / (sinΒ² π) dπ
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8. Definite Integrals
Introduction to Definite Integrals
1:48 minutesProblem 5.R.18
Textbook Question
Properties of integrals Suppose β«ββ΄ Ζ(π) dπ = 6 , β«ββ΄ g(π) dπ = 4 and β«ββ΄ Ζ(π) dπ = 2 . Evaluate the following integrals or state that there is not enough information.
ββ«βΒΉ 2Ζ(π) dπ
Verified step by step guidance1
Step 1: Recognize that the integral β«βΒΉ 2Ζ(π) dπ involves reversing the limits of integration. When the limits are reversed, the integral changes sign. Thus, β«βΒΉ 2Ζ(π) dπ = -β«ββ΄ 2Ζ(π) dπ.
Step 2: Use the property of integrals that allows constants to be factored out. Specifically, β«ββ΄ 2Ζ(π) dπ = 2β«ββ΄ Ζ(π) dπ.
Step 3: Substitute the given value of β«ββ΄ Ζ(π) dπ = 6 into the equation from Step 2. This gives β«ββ΄ 2Ζ(π) dπ = 2 Γ 6.
Step 4: Combine the results from Step 1 and Step 3 to express the integral as -β«ββ΄ 2Ζ(π) dπ = -(2 Γ 6).
Step 5: Conclude that the integral β«βΒΉ 2Ζ(π) dπ can be evaluated using the steps above, but the final numerical result is not calculated here as per the instructions.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Properties of Definite Integrals
Definite integrals have several key properties, including linearity and the ability to reverse limits. The linearity property states that β«[a,b] (c * f(x)) dx = c * β«[a,b] f(x) dx for any constant c. Additionally, reversing the limits of integration changes the sign: β«[b,a] f(x) dx = -β«[a,b] f(x) dx. Understanding these properties is essential for evaluating integrals efficiently.
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Substitution is a technique used in integration to simplify the process of evaluating integrals. It involves changing the variable of integration to make the integral easier to solve. For example, if we let u = g(x), then the integral β« f(g(x)) g'(x) dx can be transformed into β« f(u) du, which may be simpler to evaluate. This concept is particularly useful when dealing with composite functions.
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