Important Indefinite and Definite Integrals

$$\int_0^\infty e^{- \alpha^2 x}dx = \frac{1}{\alpha^2}$$

$$\int_0^\infty e^{-\alpha^2 x^2}dx = \frac{1}{2} \sqrt{\frac{\pi}{\alpha^2}}$$

$$\int \sin^2 x dx = \int \left ( \frac{e^{\imath x} - e^{-\imath x} }{2 \imath} \right )^2 dx$$