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

110

THE DEFINITE INTEGRAL

Then (x) (x) dcv = (b(a + o) (x) d. + (c )J t(x) dx,
J( Jea J
where a __ _ oo.*
Suppose ~((x) to be monotonic increasing.
We apply the Second Theorem of Mean Value to the arbitrary interval
(a, b).
Then we have
J )+(x=) (x) dr= ((a +o)J r() dx+ (b -O) (.),
Ja ja Ja
where aC -- b.
Add to both sides B= (oo ) r(x) dx,
Jb
observing that sb(ao) exists, since ((x) is monotonic increasing in x a and
does not exceed some definite number (~ 34).
Also Lt B=0 and S(x),(x) dx converges [~ 56, II.].
b->oo a
Then B+ \ (x) ~(x) dx
Ja
=+(a+)J I (x) dx+ (b - 0o)J i(x) dx~+ ()c J (x) dx
=c(a+o0)L f +(2)dx - Jji(x) d] ~1+() - 0)LVf i3(x) dx - J (x)dx1
+4(oo )J| 4()dx
a rZ~~~~~~~~~b
()d.,....................................................(1)
Ja
where U={ (b - 0) - 4(a + O)}J (x) dr,
and V={J(o )- 1(b - )}J i()d.
Now we know from the above Lemma that J l(x) dx is bounded in (a, o ).
Jx?
Let Mi, m be its upper and lower bounds.
Then m (. )dx M,
and | JV(x)dVTl.
b
Therefore {((b -0)- q(c + 0)}m U {(b - 0)- q (a + )}Jf,
{(oo) -(b-O)}2< T_ { ( oo)l-. (bI-O)}M.
*Cf. Pierpont, loc. cit., Vol. I., ~ 654.