(^228) Integrals
Problems 4.39
(^1) Make a small table of integrals by copying formulas from the second
column of (4.171) to (4.175). Combine the processes of learning these formulas
and using them to show that
(a) foI (x - x2) dx ='- (b) fo1 x(1 - x) dx
{c) foi x2(1 - x2) dx (d) f1 x2(1 - x)2 dx = sa
(e) fix dt = log x (f) jo 1 -}- x2dx = log 5
(g) fo, sin x dx = 2 (h) fog cos x dx = 0
(i) foe--dx = 1 - e-1 (j)
f'2
(x + x)2 dx =
1 h sin (a + bh) - sin a
(k) hfo cos (a + bt) dt = bh
2 Verify the formula
fo
1 plot
x'(1 - x)Qdx=
(p+q+1)i
for some pairs of small nonnegative integers p and q. Remark: Anyone who
wishes to augment his corpus of scientific information should be informed that
this is a famous and important formula. The integral is the beta integral. The
formula is correct whenever p and q are real or complex numbers with nonnega-
tive real parts. When Cauchy extensions of Riemann integrals have been defined
and are used, it can be proved that the formula is correct when p and q are com-
plex numbers with real parts exceeding -1.
3 While we are not now indulging in proofs of such things, the two integrals
(1) o (1 + x)8 1 +x
I1 1
dx,
f1 1
dx
are nearly equal when s is near 1. Nevertheless, we must use different integration
formulas to evaluate the integrals. Obey the rules and show that
(2(
(^11) _21-8-1
d
f
)) x 1 - s l
o (I + x)e
(s )(^3 ) fo 1+xdx =log 2.
Remark: While the details need not be fully understood at the present time, we
pause to learn that the right member of (2) really is near log 2 when s is near 1.
This means that(4) lim21-8 - 1= log 2.e-.1 1 - s
To see that (4) is correct, we can let(5) f(x) = exIog2