and integrate from 0 to a 1, we find

x-1

-1

(1 + x) log x

=

- log

Now by the formulæ expressing the sine and cosine of an angle in products of factors, we have

Kummer in Crelle's Journal, Vol. xvII. p. 224.

(8) By integration by parts it is found that

fdxe cosrx = − €'

a cos ra r sin ræ

a2 + p2

If we differentiate these expressions (n - 1) times with respect to a we have by Ex. (20) and (21) of Chap. 11. Sect. 1 of the Diff. Calc.

In these expressions n must be a positive integer; but if it be a positive fraction, the only difference is that instead of the continued product 1.2.3... (n − 1) we must substitute the definite integral I (n).

If we integrate (a) with respect to r, we have

no constant being added, as the integral vanishes when r = 0.

In this formula if we make a =

(ƒ)

0, we have

If we make a = 0 in the formulæ (a) and (b) we have

(g)

f* dx cos rx = 0, (h) fodx sin ræ =

From the integral (ƒ) it is easy to see that

(k)

when lies between

other values of r.

- 1 and +1, but that it vanishes for all

The results (g) and (h) are very remarkable as giving the real values of what are apparently indeterminate quantities, the sines and cosines of an infinite angle. For as

so that both the sine and the cosine of an infinite angle are equal to zero.

In the formulæ (c) and (d) if we make a = 0, we find the two remarkable integrals

If the index n lie between 0 and 1 the corresponding formulæ may be deduced without the consideration of limits involved in making a = 0. Since

fo da a ̄"e"* = T (1 − n) x2−1,

on multiplying both sides of this equation by cos rada and integrating from 0 to co, we have

fo dx cos rx f da a ̄" € ̃"" = г (1 − n) [ dx x-1 cos rx.

By the formula (d) in Ex. (5), we have

Thus if n = 1, we have, since (1) = π3, and

Г (п)

sin n

x"

fdxx-2 e sin ra when n< 1.

For, on integration by parts, the integrated term vanishes at both limits, and we have

If in formulæ (1) and (m) we assume x = ≈

f*d≈ cos (rx") = n I (n)

π

COS n

n

π

cos n

they become

Hence if n =

(g)

["dx cos r** = § (—)* = ƒ"dx sin re2.

The formulæ in this article are due principally to Euler, Calc. Integ. Vol. IV. p. 337. See also Mascheroni, Adnotationes, p. 53. Laplace, Jour. de l'Ecole Polyt. Cah. xv. p. 248, and Plana, Mémoires de Bruxelles, Vol. x.

(9) To find the value of u = ƒ”da e ̄(~+~) ̧
Differentiating with regard to a we find

.

this is a linear equation, the integral of which is

u = Ce-la.

To determine the arbitrary constant, make a = 0, when

This integral was first given by Laplace, Mémoires de l'Institut, 1810.