Textbook Question
Symmetry in integrals Use symmetry to evaluate the following integrals.
β«Β²ββ [(xΒ³ β 4x) / (xΒ² + 1)] dx
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8. Definite Integrals
Introduction to Definite Integrals
2:20 minutesProblem 5.2.53c
Textbook Question
Properties of integrals Suppose β«βΒ³Ζ(π) dπ = 2 , β«ββΆΖ(π) dπ = β5 , and β«ββΆg(π) dπ = 1. Evaluate the following integrals.
(c) β«ββΆ (3Ζ(π) β g(π)) dπ
Verified step by step guidance1
Step 1: Recognize that the integral β«ββΆ (3Ζ(π) β g(π)) dπ can be split into separate integrals using the linearity property of integrals. This property states that β«βα΅ [cβf(π) + cβg(π)] dπ = cββ«βα΅ f(π) dπ + cββ«βα΅ g(π) dπ.
Step 2: Apply the linearity property to rewrite the integral as β«ββΆ (3Ζ(π)) dπ β β«ββΆ g(π) dπ.
Step 3: Factor out the constant 3 from the first integral using the constant multiple rule, which states that β«βα΅ cΒ·f(π) dπ = cΒ·β«βα΅ f(π) dπ. This gives 3β«ββΆ Ζ(π) dπ β β«ββΆ g(π) dπ.
Step 4: Substitute the given values for the integrals. From the problem, β«ββΆ Ζ(π) dπ = β5 and β«ββΆ g(π) dπ = 1.
Step 5: Combine the results algebraically to evaluate the expression. The final result will be 3(β5) β 1.
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2mKey Concepts
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