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Assignment 6-
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 = ½_{0}∫^{2}f(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}(x^{2} + cos(x))dx
(b) _{0}∫^{2π}sin(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 sin^{3}(x). Now express sin^{3}(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 + t^{2})^{-3} dt.
(b) _{0}∫^{x^2}(1 + t^{2})^{-3} dt.
(c) _{x^2}∫^{x^3} (1 + t^{2})^{-3} dt.