An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

278 Examples on the calculation of definite integrals.

Bk. III. ch. VII.

When n is an odd integer = 2m + 1:,7t
jn2n+1 2 n 2m - 2 4 2
fsinSm^+xc = _______ --
2m + -1 2m- 1 5 3
0
Now sin x between the limits 0 and - z is a positive proper fraction,
hence:
42^t ^,n 7^
J s2m-1X (xdx > f sin2.nxdx > ji12m+-LXdX,
0 0O
2 4 2 m -2 1 3 2 -- 1 2 4 2 m- 2 2 m
3 5 'i' 2m- 1 2 ' 2 4 n 3 5 2 m - 1 2 iF 1+
Dividing across by the coefficient of z r, we find:
2 2 4 4 n 2m - 2 2 m- 2. 2
1 3 3 5 2 m - 3 2m - 1 2m - 1 2
2 2 4 4 2? -- 2 2 mn - 2 2m 2m
1 3 3 5 2 n -- 3 2 z --- 1 2m-i 1 2m-T 1
Am>n, > > m '2+t l'
A zL> 2 1 2, + I
As m increases, the quantities Am form a series of decreasing numbers
greater than tz, they must therefore have a definite limit. But on
the other hand we can choose m so large that Am and
2 In
AnL. 2m+
may differ inappreciably, therefore the limiting value of Am, differs
inappreciably from r z, or we have:
7r 2 2 4 4 6 6
22 2 T 4y3 4 -6 -51 " 17 in infinitum *).
2 1 3 3 5 5 7
We have also:
'7t!zt
cosfxdx nsinx d x.
0 0
158. Fifth group.
The following process serves for the evaluation of the integral
e-xdx.
Let us denote its value by A and introduce a new variable s by the
equation x = a, then:
*) Wallis: Arithmetica infinitorum. The earliest expression of a number
in the form of an infinite product. (Cf. ~ 39 and ~ 163.)