1a) Assume that f(x) is continuous on [1,1]. By making the substitution u = Pi - x, show that0Pixf(sinx)dx=(Pi/2)0Pif(sinx)dx

1b) Use your answer to part 1a) to evaluate the integral0Pi1+cos2xxsinxdx

1c) Let I(k) =0kx4+k4dx Show that I(k)=(1/k)I(1) for k=0

1d) Suppose that f is a continuous function satisfying0bf(ax)dx+0af(bx)dx=c for some constants a,b,c with a not equal to 0 and b not equal to zero. Let I = 0abf(x)dx. Determine I in terms of c if a = 2, b=4. Also determine I in terms of c if a = 3 and b = 6.

1e)LetIn=01xnexdx where n = 0,1,2,.......

(i) Evaluate I0.

(ii) Show that 0 < I_n< 1/(n+1)

(iii)Use (ii) to deducelimnIn

(iv) Show that I_n=nI_n-1 - 1/e