A course of pure mathematics, by G.H. Hardy.

340

MISCELLANEOUS EXAMPLES ON CHAPTER VIII

28. Establish the formulae
F{v(x2 + l)+x.} dx= (i+-) F (y)dy,
fFj{ (x2+l )-x}dx=1 (1+) F 1 ) dy,
where F is such a function that the integrals have a meaning. In particular,
prove that if n > 1 then
00 dx 0 n
o {(x+l)+x}"=o {(:2-+l)-1x} -X=n2
[First put x=sinh u and then eU=y.]
29. Show that if 2/=ax- (b/x), where a and b are positive, y increases
steadily from - oo to + oo as x increases from 0 to +oo. Hence show that
f ax +a $)}d= ft- f. %2( + ab) {+ b)}dy
= f /{(y2+ ab)}dy.
ao 0
30. Show that if 2y=ax+ (b/x), where a and b are positive, two values
of x correspond to any value of y greater than J(ab). Denoting the greater
of these by xl and the less by x2, show that, as y increases from J/(ab)
towards oo, xl increases from J/(b/a) towards oo, and x2 decreases from
V(bla) to 0. Hence show that
/(y) dx& = 1 Y f + i d
Y/(b/a) a (ab) ( ( 2 ab) + l d
r\I(bla) r( v
O(ba) a \/(ab) {vS(y2 ) }d Jb)
and that
~lo~f oyb xl 2 0 /f() 2 2 )
f{ ( ax X dx - y( ) dy = _ f f f{/(2 + ab)} dz,
fo 0{ \ x+)} J J =f(ab) (y2 - ab) a
f denoting any function such that these integrals have a meaning in accordance with the definitions of ~~ 160 et seq.
31. Prove the formula
f (sec xl + tan 1 x) 2 J -f(cosec x) $c'
() 2 (sin x) xo (sin X)
32. If a and b are positive, then
fo dx _r- b ' x2dx rr
Jo (x2+a2) (X2+b2)= 2ab (a+b)' o (x2+a2) (x2+ b2) 2 (a+ b)
Deduce that if a, 3, and y are positive, and 2> ay, then
f/ dx- _ _ r f x2dx _r
XJo a4+2,3x2+ 2 /(2yA)' J ax4 + 2x- 2 +y 2 ^(2aA)'
where A=/3+V/(ay). Also deduce the last result from Ex. 29, by putting
f(y)= l/(C2+y2). The last two results remain true when 12<ay, but their
proof is then not quite so simple.