Answer to Question #177952 in Real Analysis for Izza Eeman

Question #177952

Suppose that 𝛽 > 0 and that β„Ž ∈ 𝑅[βˆ’π›½, 𝛽]. 1) If β„Ž is even, show that t ∫ h(integration limit from [βˆ’π›½, 𝛽]) = 2 ∫ h(integration limit from [0, 𝛽])


1
Expert's answer
2021-04-15T07:39:34-0400

We should prove that for even function h

"\\int\\limits_{-\\beta}^{\\beta} h(t)\\,dt= 2\\int \\limits_{0}^{\\beta} h(t)\\,dt."

We know that h(-x)=h(x).


Let us divide the integral into two parts:

"\\int\\limits_{-\\beta}^{\\beta} h(t)\\,dt= \\int\\limits_{-\\beta}^{0} h(t)\\,dt + \\int\\limits_{0}^{\\beta} h(t)\\,dt" .

We may see that the first integral may be transformed

"\\int\\limits_{-\\beta}^{0} h(t)\\,dt = \\Big| x = -t \\Big| = -\\int\\limits_{\\beta}^{0} h(-x)\\,dx = -\\int\\limits_{\\beta}^{0} h(x)\\,dx" because h(-x)=h(x).

And "-\\int\\limits_{\\beta}^{0} h(x)\\,dx = \\int\\limits_{0}^{\\beta} h(x)\\,dx" .

Therefore, "\\int\\limits_{-\\beta}^{\\beta} h(t)\\,dt= \\int\\limits_{0}^{\\beta} h(x)\\,dx + \\int\\limits_{0}^{\\beta} h(t)\\,dt = 2\\int\\limits_{0}^{\\beta} h(t)\\,dt" .


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment

LATEST TUTORIALS
New on Blog
APPROVED BY CLIENTS