Introduction to the theory of Fourier's series and integrals, by H.S. Carslaw.

102 THE DEFINITE INTEGRAL
For from ~ 47 we know that
g(x) dxJ f(x) dx, when x>a.
Therefore Jy (x) dx < f(x) dx.
a ia
Then, from I., g(x) d x f(x) dx.
III. If (g(x) f(x), and f(x) dx diverges, so also does
co
g (x) dx.
a
This follows at once, since g(x)dx f(x)dx.
a a
One of the most useful integrals for comparison is | dx, where
a>O. a
We have J =x -n {l x1-a1-}, when n+1,
(' x dx
and I = logx-loga, when n =1.
a
Thus, when n>], Lt xd: 'X C
— >z.JaX^ n-I
i.e. |dx alJ 1 zX
And, when t — l, Lt
xLt 't X ~
i.e. Idx diverges.
a X'
' Since the relative behaviour of the positive integrands f(x) and g(x) matters
only as x —>o, these conditions may be expressed in terms of limits:
When g(x)/f(x) has a limit as x->, | g(x) dx converges, if ff(x) dx converges.
When g(x)/f(x) has a limit, not zero, or diverges, as x->oc, g(x) dx diverges,
aif (x)x ges.