1. For any real number x, [x] be the greatest integer less than or equal to x. For example [3.21] = 3 and [4.95] = 4. Determine each of the following integrals. Explain your answer.
(a) 2∫10 [x] dx.
(b) 4∫64 [√x] dx.
2. Prove that if f(x) is a step function defined on the interval [0, 1], then
0∫1 f(x) dx = ½0∫2f(x/2) dx.
3. Determine whether or not the following functions are integrable on the interval [0, 1].
(a) f(x) where f(x) = x if x ∈ Q and f(x) = 0 if x ∉ Q.
4. Compute the following integrals. Explain your answers.
(a) 0∫π/2(x2 + cos(x))dx
(b) 0∫2πsin(2x) cos(x) dx,
without using integration by parts, or any substitution beyond ones of the form u = cx for some constant c. (Hint: Express the integrand in terms of sin(x) and sin3(x). Now express sin3(x) in terms of sin(x) and sin(3x).)
5. Without attempting to evaluate the following indefinite integrals, find f'(x) in each case if f(x) is equal to
(a) 0∫x(1 + t2)-3 dt.
(b) 0∫x^2(1 + t2)-3 dt.
(c) x^2∫x^3 (1 + t2)-3 dt.