Laplace integral

An integral of the form

$$\int\limits_0^\infty f(t)e^{-pt}dt\equiv F(p),$$

that defines the integral Laplace transform of a function $f(t)$ of a real variable $t$, $0<t<\infty$, giving a function $F(p)$ of a complex variable $p$. It was considered by P. Laplace at the end of the eighteenth and beginning of the 19th century; it was used by L. Euler in 1737.

Two specific definite integrals depending on the parameters $\alpha,\beta>0$:

$$\int\limits_0^\infty\frac{\cos\beta x}{\alpha^2+x^2}dx=\frac{\pi}{2\alpha}e^{-\alpha\beta},$$

$$\int\limits_0^\infty\frac{x\sin\beta x}{\alpha^2+x^2}dx=\frac\pi2e^{-\alpha\beta}.$$

References

[a1]	F. Oberhettinger, L. Badii, "Tables of Laplace transforms" , Springer (1973)
[a2]	I.N. Sneddon, "The use of integral transforms" , McGraw-Hill (1972) pp. Chapt. 6
[a3]	V.A. Ditkin, A.P. Prudnikov, "Integral transforms" , Plenum (1969) (Translated from Russian)
[a4]	G. Doetsch, "Handbuch der Laplace-Transformation" , 1–3 , Birkhäuser (1950–1956)