Math 1B
§ 5.4 Indefinite Integrals
Overview: Here we will cover notation for antiderivatives, review the formulas for antiderivatives, and
evaluate indefinite integrals.
The Indefinite Integral
The set of all antiderivatives of the function f is called the indefinite integral of f with respect to
x, and is symbolized by
∫ f ( x )dx
Notice, there are no limits of integration in an indefinite integral.
€ constant C to an indefinite integral,
We always add the arbitrary
because it represents an entire family of functions. The figure
to the right represents the family of curves y = x3 + C, which fill
the coordinate plane without overlapping. Remember, C can be
any real number.
So, when we say that
€
∫
∫ f ( x )dx = F ( x ) we mean that F "( x ) = f ( x ) . For example,
x4
+C
4
x 3 dx =
%
d " x4
3
$ + C' = x .
dx # 4
&
because
€
Notice that the indefinite integral ∫ f ( x ) dx represents a family of functions (that differ by a constant,
b
€
f ( x ) dx , is a number.
C), whereas
a definite integral, (that has limits€of integration),
∫
a
Also, if f is a continuous€
function, the definite integral is the indefinite integral evaluated on the
interval [a, b], that is,
b
b
a
a
∫ f ( x )dx = ∫ f ( x )dx ]
.
€
Table of Indefinite Integrals:
1. On the next page is our table of indefinite integrals. We adopt the convention that when a formula
€
for a general indefinite integral is given, it is valid only on an interval.
For example, it is understood that the formula
1
∫x
2
1
dx = − + C is only valid on the interval (0,∞)
x
or on the interval (−∞,0) .
2. To evaluate integrals in 5.3 & 5.4, it is easiest to write out the integrand as a sum or€
difference of
€
terms that can be integrated using the table on the following page.
€
Stewart – 7e
1
Table of Indefinite Integrals
∫ c f (x ) d x
∫k
=c
∫ f (x ) d x
∫ !# f (x ) + g (x )"$ d x = ∫ f (x ) d x + ∫ g (x ) d x
dx = kx + C
n
∫ x dx =
x n +1
+C
n +1
(n
≠ − 1)
∫x
−1
1
dx =
∫x
ax
+C
lna
x
x
∫ e dx = e + C
x
∫ a dx =
∫ sin x dx
∫ cos x dx
∫ sec
2
= − cos x + C
x dx = tan x + C
∫ sec x tan x dx
∫x
2
= sec x + C
1
d x = t a n− 1 x + C
+1
∫ sinh x dx
= cosh x + C
∫ csc
2
dx = ln x + C
= sin x + C
x dx = − cot x + C
∫ csc x cot x dx
∫
1
1− x
2
= − csc x + C
d x = s i n− 1 x + C
∫ cosh x dx
= sinh x + C
Example: Find the general indefinite integral.
a)
∫ sec x (sec x + tan x )dx
€
b)
∫(
d)
∫
)
t 3 + 3 t 2 dt
€
c)
#
∫ %$ x
2
−5+
&
(dx
1− x 2 '
1
sin2x
dx
sin x
€
€
Stewart – 7e
2
Example: Evaluate each integral.
a)
∫
9
0
2u du
€
b)
∫
d)
∫
y−y
dy
y2
4
1
€
c)
5
∫ (2e
0
€
x
+ 4 sin x )dx
1+ sin 2 x
dx
π
4
sin 2 x
π
2
€
Stewart – 7e
3