# §6.14 Integrals

## §6.14(i) Laplace Transforms

 6.14.1 $\int_{0}^{\infty}e^{-at}E_{1}\left(t\right)\,\mathrm{d}t=\frac{1}{a}\ln\left(1% +a\right),$ $\Re a>-1$,
 6.14.2 $\int_{0}^{\infty}e^{-at}\operatorname{Ci}\left(t\right)\,\mathrm{d}t=-\frac{1}% {2a}\ln\left(1+a^{2}\right),$ $\Re a>0$,
 6.14.3 $\int_{0}^{\infty}e^{-at}\operatorname{si}\left(t\right)\,\mathrm{d}t=-\frac{1}% {a}\operatorname{arctan}a,$ $\Re a>0$.

## §6.14(ii) Other Integrals

 6.14.4 $\int_{0}^{\infty}{E_{1}}^{2}\left(t\right)\,\mathrm{d}t=2\ln 2,$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential of $x$, $E_{1}\left(\NVar{z}\right)$: exponential integral, $\int$: integral and $\ln\NVar{z}$: principal branch of logarithm function A&S Ref: 5.1.33 Permalink: http://dlmf.nist.gov/6.14.E4 Encodings: TeX, pMML, png See also: Annotations for §6.14(ii), §6.14 and Ch.6
 6.14.5 $\int_{0}^{\infty}\cos t\operatorname{Ci}\left(t\right)\,\mathrm{d}t=\int_{0}^{% \infty}\sin t\operatorname{si}\left(t\right)\,\mathrm{d}t=-\tfrac{1}{4}\pi,$
 6.14.6 $\int_{0}^{\infty}{\operatorname{Ci}}^{2}\left(t\right)\,\mathrm{d}t=\int_{0}^{% \infty}{\operatorname{si}}^{2}\left(t\right)\,\mathrm{d}t=\tfrac{1}{2}\pi,$
 6.14.7 $\int_{0}^{\infty}\operatorname{Ci}\left(t\right)\operatorname{si}\left(t\right% )\,\mathrm{d}t=\ln 2.$

## §6.14(iii) Compendia

For collections of integrals, see Apelblat (1983, pp. 110–123), Bierens de Haan (1939, pp. 373–374, 409, 479, 571–572, 637, 664–673, 680–682, 685–697), Erdélyi et al. (1954a, vol. 1, pp. 40–42, 96–98, 177–178, 325), Geller and Ng (1969), Gradshteyn and Ryzhik (2000, §§5.2–5.3 and 6.2–6.27), Marichev (1983, pp. 182–184), Nielsen (1906b), Oberhettinger (1974, pp. 139–141), Oberhettinger (1990, pp. 53–55 and 158–160), Oberhettinger and Badii (1973, pp. 172–179), Prudnikov et al. (1986b, vol. 2, pp. 24–29 and 64–92), Prudnikov et al. (1992a, §§3.4–3.6), Prudnikov et al. (1992b, §§3.4–3.6), and Watrasiewicz (1967).