sequences and series - The equivalence of an integral and a sum of integrals

Let $p(x)$ be a polynomial.

Assume that $ \displaystyle \int_{a}^{b} p(x) \cot \left(\frac{ax}{2} \right) \ dx $ converges.

Then $$ \int_{a}^{b} p(x) \cot \left(\frac{ax}{2} \right) \ dx = 2 \sum_{n=0}^{\infty} \int_{a}^{b} p(x) \sin(anx) \ dx. $$

I can verify that this formula is true in particular cases, but I'm not sure how to go about proving it.

EDIT:

The lower limit and the integrand parameter don't need to be the same.

So the identity could be written as $$ \int_{a}^{b} p(x) \cot \left(\frac{ r x}{2} \right) \ dx = 2 \sum_{n=0}^{\infty} \int_{a}^{b} p(x) \sin( r nx) \ dx .$$

And as was mentioned below, $p(x)$ need not be a polynomial.

There are three other similar identities.

They are

$$ \int_{a}^{b} p(x) \tan \left(\frac{rx}{2} \right) \ dx = -2 \sum_{n=0}^{\infty} (-1)^{k} \int_{a}^{b} p(x) \sin(rnx) \ dx ,$$

$$\int_{a}^{b} p(x) \csc \left(rx \right) \ dx = 2 \sum_{n=0}^{\infty} \int_{a}^{b} p(x) \sin[(2n+1)rx] \ dx, $$

and
$$ \int_{a}^{b} p(x) \sec \left(rx \right) \ dx = 2 \sum_{n=0}^{\infty} (-1)^{k} \int_{a}^{b} p(x) \cos[(2n+1)rx] \ dx. $$

They all can be derived in a manner similar to way Daniel Fischer derived the first one by using the identities

$$\sum_{n=0}^{N} (-1)^{n} \sin(rnx) = - \frac{1}{2} \tan \left(\frac{rx}{2}\right) + \frac{(-1)^{n} \sin [(N+\frac{1}{2})rx]}{2\cos (\frac{rx}{2})}, $$

$$\sum_{n=0}^{N} \sin [(2n+1)rx] = \frac{1}{2} \csc (rx) - \frac{\cos [2(N+1)rx]}{2 \sin (rx)}, $$

and

$$ \sum_{n=0}^{N} (-1)^{n} \cos [(2n+1)rx] = \frac{1}{2} \sec(rx) + \frac{(-1)^{n}\cos [2(N+1)rx]}{2 \cos (rx)}$$

respectively.

Answer

Basically, because of the Riemann-Lebesgue lemma. By summing a geometric sum, or by induction using trigonometric identities, one finds

$$\sum_{n=0}^N 2\sin (anx) = \cot \frac{ax}{2} - \frac{\cos \left(a(N+\frac12)x\right)}{\sin \frac{ax}{2}}.$$

So that yields

$$\int_a^b p(x) \cot \frac{ax}{2}\,dx = 2\sum_{n=0}^N \int_a^b p(x)\sin (anx)\,dx + \int_a^b p(x)\frac{\cos \left(a(N+\frac12)x\right)}{\sin \frac{ax}{2}}\,dx.$$

Now if $\int_a^b p(x)\cot \frac{ax}{2}\,dx$ converges, the same is true for

$$\begin{align}
\int_a^b p(x)\frac{\cos \left(a(N+\frac12)x\right)}{\sin \frac{ax}{2}}
&= \int_a^b p(x) \frac{\cos (aNx)\cos \frac{ax}{2} - \sin (aNx)\sin \frac{ax}{2}}{\sin \frac{ax}{2}}\,dx\\
&= \int_a^b p(x) \cot \frac{ax}{2}\cos (aNx)\,dx - \int_a^b p(x)\sin (aNx)\,dx,
\end{align}$$

and by the Riemann-Lebesgue lemma, both of these integrals converge to $0$ for $N \to \infty$.