Example 7.4.7(a): Riemann implies Lebesgue Integrable
Let g'(x) = cos(x), f(x) = x and G(x) = x sin(x). Using integration by parts with a = -1 and b = 1 we have:
x cos(x) = G(b) - G(a) - sin(x) =Therefore the Lebesgue integral
= b sin(b) - a sin(a) + (cos(b) - cos(a)) =
= sin(1) + sin(-1) + (cos(1) - cos(-1)) = 0
[-1, 1] x cos(x) = 0as well.
Note that we have shown in a prior example that the (Riemann) integral of an odd function over an interval [-a, a] is always zero. Since x cos(x) is odd, the integral must therefore be zero by that example.