James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 5.4 Indefinite Integrals and the Net Change Theorem 403
Theorem, the notation y f sxd dx is traditionally used for an antiderivative of f and is
called an indefinite integral. Thus
y f sxd dx − Fsxd means F9sxd − f sxd
For example, we can write
y x 2 dx − x 3
3 1 C because d dx
S x 3
3 1 C D − x 2
So we can regard an indefinite integral as representing an entire family of functions (one
antiderivative for each value of the constant C).
You should distinguish carefully between definite and indefinite integrals. A definite
integral y b a
f sxd dx is a number, whereas an indefinite integral y f sxd dx is a func tion (or
family of functions). The connection between them is given by Part 2 of the Fundamental
Theorem: If f is continuous on fa, bg, then
y b
b
f sxd dx − y f sxd dxg
a
a
The effectiveness of the Fundamental Theorem depends on having a supply of antiderivatives
of functions. We therefore restate the Table of Antidifferentiation Formulas
from Section 4.9, together with a few others, in the notation of indefinite integrals. Any
formula can be verified by differentiating the function on the right side and ob tain ing the
integrand. For instance,
d
y sec 2 x dx − tan x 1 C because
dx stan x 1 Cd − sec2 x
1 Table of
Indefinite Integrals
y cf sxd dx − c y f sxd dx
y k dx − kx 1 C
y x n dx −
x n11
1 C sn ± 21d y
n 1 1
y f f sxd 1 tsxdg dx − y f sxd dx 1 y tsxd dx
1
x dx − ln | x | 1 C
y e x dx − e x 1 C y b x dx − b x
ln b 1 C
y sin x dx − 2cos x 1 C
y sec 2 x dx − tan x 1 C
y sec x tan x dx − sec x 1 C
1
y
x 2 1 1 dx − tan21 x 1 C
y sinh x dx − cosh x 1 C
y cos x dx − sin x 1 C
y csc 2 x dx − 2cot x 1 C
y csc x cot x dx − 2csc x 1 C
1
y dx − sin 21 x 1 C
s1 2 x 2
y cosh x dx − sinh x 1 C
Recall from Theorem 4.9.1 that the most general antiderivative on a given interval is
obtained by adding a constant to a particular antiderivative. We adopt the convention
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