Mathematical Methods for Physics and Engineering - Matematica.NET

2.4 HINTS AND ANSWERS2.45 If J r is the integral∫ ∞0x r exp(−x 2 ) dxshow that(a) J 2r+1 =(r!)/2,(b) J 2r =2 −r (2r − 1)(2r − 3) ···(5)(3)(1) J 0 .2.46 Find positive constants a, b such that ax ≤ sin x ≤ bx for 0 ≤ x ≤ π/2. Usethis inequality to find (to two significant figures) upper and lower bounds for theintegral∫ π/2I = (1+sinx) 1/2 dx.0Use the substitution t =tan(x/2) to evaluate I exactly.2.47 By noting that for 0 ≤ η ≤ 1, η 1/2 ≥ η 3/4 ≥ η, prove that23 ≤ 1 ∫ a(a 2 − x 2 ) 3/4 dx ≤ π a 5/2 04 .2.48 Show that the total length of the astroid x 2/3 + y 2/3 = a 2/3 , which can beparameterised as x = a cos 3 θ, y = a sin 3 θ,is6a.2.49 By noting that sinh x< 1 2 ex < cosh x, andthat1+z 2 < (1 + z) 2 for z>0, showthat, for x>0, the length L of the curve y = 1 2 ex measured from the originsatisfies the inequalities sinh x