563489578934

A–4 Integral Calculus 671
d sin ax
dx
d cos ax
dx
d tan ax
dx
d sin -1 ax
dx
b(x)
dc f(l, x) dl d
L
a(x)
= a cos ax
= -a sin ax
dx
= f(b(x), x) db(x)
dx
d cos -1 ax
dx
d[e ax ]
dx
a
d[a x ]
=
cos 2 ax
dx
= a x ln a
a
d(ln x)
=
31 - (ax) 2 dx
= 1 x
d(log a x)
dx
= 1 x log ae
- f(a(x), x) da(x)
dx
a
= -
31 - (ax) 2
= ae ax
+
L
b(x)
a(x)
0f (l, x)
0x
dl
(Leibniz’s rule)
A–3 INDETERMINATE FORMS
If lim x:a f(x) is of the form
then
0
0 , q
q , 0 # q, q-q, 0°, q°, 1 q
lim f(x) = lim c N(x)
x:a x:a D(x) d = lim c (dN(x)>dx)
d (L’HoNpital’s rule)
x:a (dD(x)>dx)
where N(x) is the numerator of f(x), D(x) is the denominator of f(x), N(a) = 0, and D(a) = 0.
A–4 INTEGRAL CALCULUS
Definition
f(x) dx = lim
L e ¢x:0
a [f(n ¢x)]¢ xf
n