## Example 7.2.7: Integration by Parts and Limits

Suppose

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

Thus,

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