Question 5. Let A be the area enclosed between p(x)=1 V2-sin x , the x-axis and two vertical lines x= and X=> 2 ; let B be the area enclosed between q(x)=- 1 V2+sin x , the x -axis and two lines x= = 2 and x3 x 2 Show that A = B (the two enclosed area are equal). // Hints: areas can be expressed as some definite integrals. // Hints: to prove A = B, you can do one of the following: I. Calculate the results of A, B and show they are equal; II. Show A - B = 0; III. Show the expression of A can be transformed into B through algebraically equivalent transforms. Question 6. * exp(-x) denote the exponential function e -x) ; e=2.718 ... is the base for natural logarithm. For example, at x = 2, exp(-22) = e(-2 ) = e-4 // In particular, the function g(x)=exp(-x2) is shown as in the right figure. 0.6 (i) Find the maximum value for function f(x) = x2p-1exp(-x ) , where p20 is a parameter. 0.4 0.2 At what value of x does function f(x) achieves its maximum? If p = 3, at what point is the maximum of 0.2 0 0.2 04 0.6 12 14 16 1 8 f(x)? (ii) Apply numerical integration to calculate 9/4 exp(- x- 2 3/4 dx . ( You may use 3 - 5 trapezoids for this calculation; show calculation steps). XQuestion 7. (i) Show that S ( V3+ 24 du = 2V3+2u+3 du u uv3+2u Hint: to prove that S f (x) dx = P(x) - S g(x) dx , you may want to try a proof for S (f (x) +g(x)) dx=P(x). (ii) Suppose we introduce F(u) and G(u) as: S( V3 + 24) )du = F(u), and V3+2u = G(u) u Find the integral of du u V3+ 2u in terms of a F(u) + b G(u), where a, b are two real number constants. Hint: applying the relation given in (i). (iii) Calculate the following derivative: du d (273 + 2u ) . (iv) Let a new variable x be introduced as x =3+2u . Show that =x , and dx (V3+24) du = 2 x - dx u X - 3 Please do not proceed beyond the requirement of this question. // Note: For Question 7, the Fundamental Theorem of Calculus may help you understand and validate