(xi.)
| ∫∞0 | cos ax − cos bx | dx = log | b | , |
| x | a |
(xii.)
| ∫∞0 | cos x − e−mx | dx = log m, |
| x |
(xiii.)
∫∞−∞ e−x2+2ax dx = √π · ea2,
(xiv.)
∫∞0 x−1/2 sin x dx = ∫∞0 x−1/2 cos x dx = √(½ π),
53. Multiple Integrals.The meaning of integration of a function of n variables through a domain of the same number of dimensions is explained in the article [Function]. In the case of two variables x, y we integrate a function ƒ(x, y) over an area; in the case of three variables x, y, z we integrate a function ƒ(x, y, z) through a volume. The integral of a function ƒ(x, y) over an area in the plane of (x, y) is denoted by
∫∫ ƒ(x, y) dx dy.