Answer to Question #178141 in Real Analysis for Izza Eeman

Question #178141

Suppose that 𝛽 > 0 and that β„Ž ∈ 𝑅[βˆ’π›½, 𝛽]. 1) If β„Ž is even, show that ∫ h(integration limit from [βˆ’π›½, 𝛽]) = 2 ∫ h(integration limit from [0, 𝛽])


1
Expert's answer
2021-04-15T07:40:46-0400

Given,

Let "h=f(x)"


As h is an even functioni.e. "f(x)=f(-x)"


"\\int _{-\\beta}^{\\beta}h. dx"


Since Above integral is additive in nature,

So


"\\int_{\\beta}^{\\beta}f(x)dx=\\int _{-\\beta}^{0}f(x) dx+\\int _{0}^{\\beta} f(x) dx"


Then, for the first integral we use the expansion/contraction of the interval of integration with k=-1 to get


"\\int_{-\\beta}^0f(x) dx=-\\int_{\\beta}^0f(-x)dx"


Since f(x) is an even function by assumption, we have "f(-x) = f(x)" for all "x \\in [0,b]" . Since"-\\int_b^0 = \\int_0^b" we then have,


Β "-\\int_b^0 f(-x) dx = \\int_0^b f(x) dx."


Putting this all together we have-


"\\int_{-\\beta}^{\\beta}f(x)dx=\\int _{0}^{\\beta}f(x) dx+\\int _{0}^{\\beta} f(x) dx=2\\int_0^{\\beta}f(x) dx"


Hence, "\\int_{-\\beta}^{\\beta}h=2\\int_0^{\\beta}h"


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