Evaluate the integrals - in 0intpi sinnxsinx dx and nbspjn

Question

Putnam TNG - Evaluation of Integrals

1: (a) Evaluate the integrals

I_n = ₀∫^π sin(nx)/sin(x) dx and  J_n = ₀∫^π(sin(nx)/sin(x))² dx, for n = 1, 2, 3 . . . .

(b) Evaluate the integrals

I_n = -_π∫^π (sin(nx)/(1 + 2^-x) sin(x)) dx, for n = 1, 2, 3 . . . .

2: Evaluate ₀∫^∞(arctan(πx) - arctan(x)/x)dx.

3: Let H be the unit hemisphere {(x, y, z): x² + y² + z² = 1, z ≥ 0}, C the unit circle {(x, y, 0): x² + y² = 1}, and P the regular pentagon inscribed in C. Determine the surface area of that portion of H lying over the planar region inside P, and write your answer in the form A sin α + B cos β, where A, B, α, β are real numbers.

4: Let ||u|| denote the distance from the real number u to the nearest integer (for example ||2.8|| = .2 = ||3.2||). For positive integers n, let

a_n = 1/n ₁∫ⁿ ||n/x||dx

Determine lim_n→∞ a_n. You may assume the identity

(2/1)(2/3)(4/3)(4/5)(6/5)(6/7)(8/7)(8/9)· · ·  = π/2.

5: Let f be a twice differentiable function on [0, 2] such that f(x) > 0 and f''(x) ≥ 0 for all x and

0∫1 f(t) dt · ₁∫² 1/f(t) dt ≤ 1.

Prove that

₀∫² f(t) dt ≤ 2f(2).

6: Show that the improper integral lim_{B→∞ 0}∫^B sin(x) sin(x²) dx converges.

7: Let f(x) be a continuous function on the interval I = [0, 1] with the property xf(y) + yf(x) ≤ 1

for x, y in I. Prove that

₀∫¹f(x) dx ≤ π/4

and find a function f(x) for which equality is obtained.