HP -15C Advanced Functions Handbook page 58

56 Section 2: Working With \7j]
• One or both limits of integration are ±°°, such as
/
The integrand tends to ±°° someplace in the range of
integration, such as
f
1
I ln(u) du = 1.
J
0
• The integrand oscillates infinitely rapidly somewhere in the
range of integration, such as
f
1
J cos (In u)du = Vz.
Equally troublesome are nearly improper integrals, which are
characterized by
• The integrand or its first derivative changes wildly within a
relatively narrow subinterval of the range of integration, or
oscillates frequently across that range.
The HP-15C attempts to deal with certain of the second type of
improper integral by usually not sampling the integrand at the
limits of integration.
Because improper and nearly improper integrals are not
uncommon in practice, you should recognize them and take
measures to evaluate them accurately. The following examples
illustrate techniques that are helpful.
Consider the integrand
Cl N
7-2 In cos(x
2
)
f ( x ) = .
x*
This function loses its accuracy when x becomes small. This is
caused by rounding cosO
2
) to 1, which drops information about
how small x is. But by using u = cos(x
2
), you can evaluate the
integrand as