State and prove the Dominated Convergence Theorem. [You may assume the Monotone Convergence Theorem.]

Let $a$ and $p$ be real numbers, with $a>0$. Prove carefully that

$$\int_{0}^{\infty} e^{-a x} \sin p x d x=\frac{p}{a^{2}+p^{2}}$$

[Any standard results that you use should be stated precisely.]