202021: functions, integrals. Recall that we say a function g is even if and only ifg(x) = g(−x) for all real x and -x in the domain of g. We say g is odd if and onlyif g(x) = −g(−x) for all real x and −x in the domain of g.a. Let E be an even function, and O be an odd function, both integrable on [-a, a]for a some real number. Show that ſª E(x) dx 2 E(x) dx. ComputeS²₂ O(x) dx.b. Let f be a function defined on some interval /defined by Ef(x)Of(x) = f(x)=f(-x) for all x € / is odd. What is (Eƒ + Of)(x)?(-a, a). Show that Eff(x)+f(-x) for all x € / is even, and that Of defined by22===c. Show that for any s € 1, ſå f(x) dx = §§ Eƒ(x) dx.S-SaLexad. Without computing the integral directly, use the above facts and any othersnecessary from the calculus section of MATH1005 to explain whyexp (x) dx = 2 sinh a.

Answered: Image /qna-images/answer/d63a205a-25b2-4299-888c-2bd5a3cc2e70.jpg

202021: functions, integrals. Recall that we say a function g is even if and only i g(x) = g(-x) for all real x and -x in the domain of g. We say g is odd if and only if g(x) = −g(−x) for all real x and -x in the domain of g. a. Let E be an even function, and O be an odd function, both integrable on [-a, a for a some real number. Show that E(x) dx = 2 ſª E(x) dx. Compute S2₂0(x) dx. a b. Let f be a function defined on some interval / (-a, a). Show that E- defined by Ef(x) f(x)+f(-x) for all x € / is even, and that Of defined by Of(x) = f(x)=f(-x) for all x € / is odd. What is (Ef + Of)(x)? 2 2 = = c. Show that for any s € 1, §³§ f(x) dx = §³§ Eƒ(x) dx. -S d. Without computing the integral directly, use the above facts and any others necessary from the calculus section of MATH1005 to explain why a Le exp (x) dx = 2 sinh a.

202021: functions, integrals. Recall that we say a function g is even if and only i g(x) = g(-x) for all real x and -x in the domain of g. We say g is odd if and only if g(x) = −g(−x) for all real x and -x in the domain of g. a. Let E be an even function, and O be an odd function, both integrable on [-a, a for a some real number. Show that E(x) dx = 2 ſª E(x) dx. Compute S2₂0(x) dx. a b. Let f be a function defined on some interval / (-a, a). Show that E- defined by Ef(x) f(x)+f(-x) for all x € / is even, and that Of defined by Of(x) = f(x)=f(-x) for all x € / is odd. What is (Ef + Of)(x)? 2 2 = = c. Show that for any s € 1, §³§ f(x) dx = §³§ Eƒ(x) dx. -S d. Without computing the integral directly, use the above facts and any others necessary from the calculus section of MATH1005 to explain why a Le exp (x) dx = 2 sinh a.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter5: Inverse, Exponential, And Logarithmic Functions

Section5.3: The Natural Exponential Function

Problem 52E

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