| ∫1/2π0 sin2n+1 x dx = ∫1/2π0 cos2n+1 x dx = | 2·4 ... 2n | , (n an integer). |
| 3·5 ... (2n + 1) |
(xi.)
| ∫ | dx | can be reduced by one of the substitutions |
| (1 + e cos x)n |
| cos φ = | e + cos x | , cosh u = | e + cos x | , |
| 1 + e cos x | 1 + e cos x |
of which the first or the second is to be employed according as e < or > 1.
50. New transcendents.Among the integrals of transcendental functions which lead to new transcendental functions we may notice
| ∫x0 | dx | , or ∫log x−∞ | ez | dz, |
| log x | z |
called the “logarithmic integral,” and denoted by “Li x,” also the integrals
| ∫x0 | sin x | dx and ∫x∞ | cos x | dx, |
| x | x |
called the “sine integral” and the “cosine integral,” and denoted by “Si x” and “Ci x,” also the integral