CEflTAIN APPLICATIONS TO THE THEOEY OF DEFINITE INTEGEALS. 781 
In like manner 
( 6 .) 
and similarly, 
^ cos^x^-\-^^dx=^ cos(v^-\-2a)dv 
^00 ^00 
=cos2aj cos('y®)<?y— sin sm(v^)dv 
(cos 2«!— sin 2«) 
sin (i} 2a+sin 2a), 
which are known relations. We may by the same method deduce the relations 
y dxi ^"''^‘^°'^cos|^a^+^^sin ^|=5r*£“^“‘^®"®cos ^2a sin 
d.rg~^'^'^*'^‘'“"%in|^a’^+|2^sin^|=5r^£~^“'=°"®sin^2asin^+0, 
originally given by Cauchy. All the above definite integrals are reduced by the theorem 
to immediate dependence on the fundamental theorem 
Vffl 
Again, let us consider the definite integral 
( 7 .) 
( 8 .) 
(9.) 
Making x^=z, we have 
^00 
“=^I ■ 
C” dx.x^ 
{a + bx + 
-i 
cx^Y 
dz 
(5 + «=+j)" 
dz r 
j -«= J h -i- f J- 
dv 
* + + by (4.) 
=j: 
=i 
dv 
(& + 2 v/ ac + cv^y 
dv 
(6 + 2 'V^ac + cv^)” 
Now from the known theorem 
I. 
r(«)r(r-«) 
in which 7' > a, we readily deduce 
dv.v^ 
i 
{i+ty- r(.) 
, r(»-^)r(^) 
w+ 1 m + 1 
{p + qv^)^ 2p 2^2 
r(n) 
( 10 .) 
