7.2. Integration Techniques

Example 7.2.7: Integration by Parts and Limits

Suppose f:[a, b] R is a continuously differentiable function. Show that:
  1. f(x) sin(nx) dx = 0
  2. f(x) cos(nx) dx = 0

We already mentioned that this will be an application of integration by parts. Let's focus on the first statement and let g'(x) = sin(nx), where g' is the function in the Integration by Parts theorem. Then:

f(x) sin(n x) dx = - 1/n cos(nx) f(x) + 1/n f'(x) cos(nx) dx

But |sin(nx)| 1 and |cos(nx)| 1, and since both f and f' are continuous functions on a closed, bounded interval there are constants K and L such that |f(x)| K and |f'(x)| L. Putting everything together we get:

| f(x) sin(n x) dx | K/n + L(b-a)/n


f(x) sin(n x) dx = 0
The proof of the second statement is left as an exercise.
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