Rory has been challenged by Maggie with a fiendish integral; I = f(x) da, where f(z) = eat sech(5x) for a E (-8, 8) . V64 - 22 To get an idea of what to do, Rory tries Maple to see what the answer is, but it is a bit unhelpful: > f := x -> exp (5*x) * sech (5*x) /sqrt (64-x^2) : integrate (f (x) , x=-8 . . 8) ; e5 sech (5 x) dx 1-8 V-12 + 64 > evalf ($) ; 3.141592654 This holds even when 5 in the integrand is replaced with any other real number Rory tries. For example, replacing 5 with 132 gives the following outputs: > h := x -> exp (132*x) *sech (132*x) /sqrt (64-x^2) : integrate (h (x) , *=-8. .8) ; 132~ sech (132 x ) dx V-x2 +64 > evalf ($) ; 3.141592654 In fact, no matter what software they use, Rory finds that the corresponding integral is approximately 7. Maggie suggests that f has some special property that Rory can use. (a) First, give a short proof that the function f is neither odd nor even. Essay box advice: For mathematical expressions, you can use either the equation editor or approximate Maple syntax. Your answer doesn't need accurate Maple syntax, it just needs to be clear to the marker. =: Q S Equation A - A- IX BI US X X Styles Font Size Words: 0(b) Despite this result, Rory wants to use properties of odd and even functions here. In particular, Rory wants to decompose ed into the sum of an even and an odd function; that is, e5x = h(x) + k(x), where h(-x) = h(x), k(-2) = -k(x). In fact, the unique choice for h and k that Rory has is h(ac) = Click for List k(a) = Click for List (c) Using (b) or otherwise, help Rory decompose the integrand f into f(x) = h2(x) + k2(2), where h2 is even and k2 is odd. Fully simplifying these functions gives h2(z) = k2 (z) = (d) Finally, briefly explain to Rory in words why 1= f(x)da = 8 1 dx. V64 -22 Explain why this integral has an exact value of IT. Essay box advice: In your explanation, you don't need to use exact Maple syntax or use the equation editor, as long as your expressions are sufficiently clear for the reader. For example, you can write . f(z) and h2 (x) as 'f(x)' and 'h_2(x)', respectively, 2 V3+4x as '2/sqrt(3 + 4x)', . sech (7) as 'arcsech(pi)' Q > Equation A - A-I BIUS X X Styles Font Size Words: 0