Section 6.1 Antiderivatives
Definition:
A Function F is called an antiderivative of f on an interval (a, b) if F 0 (x) = f (x) for
all x in (a, b).
Theorem:
If F is an antiderivative of f on an interval (a, b), then the most general antiderivative of
f on (a, b) is
F (x) + C
where C is an arbitrary constant.
Definition:
The
Z collection of all antiderivatives of a function f (x) is called the indefinite integral
and is denoted by f (x) dx.
If we know one function F (x) for which F 0 (x) = f (x), then
Z
f (x) dx = F (x) + C
where C is an arbitrary constant and called the constant of integration.
integration Rules:
Z
Z
kf (x) dx = k
f (x) dx (Constant Multiple Rule)
Z
Z
[f (x) ± g(x)] dx =
Z
Z
Z
xn dx =
Z
f (x) dx ±
g(x) dx (Sum/Difference Rule)
1
xn+1 + C, n 6= −1 (Power Rule)
n+1
1
dx = ln |x| + C (Indefinite Integral of x−1 = x1 )
x
ex dx = ex + C (Indefinite Integral of Exponential Function)
1. Find the most general antiderivative of the following functions. (Use C for the constant of
integration. Remember to use absolute values where appropriate.)
(a) f 0 (x) = 5x4 − 22x + 9
(b) f 0 (x) = 9x9 − 4x6 + 11x3
Z
(c)
12x + 13x11 dx
Z 1
x
−4
(d)
−e + x −
dx
2
Z
(e)
3x−3 + 4x−1 dx
2. Find the most general antiderivative of the following functions. (Use C for the constant of
integration. Remember to use absolute values where appropriate.)
Z √
(a)
5 x5 + 4ex dx
Z (b)
x2 + 7x − 4
x3
dx
2
Fall 2016,
©
Maya Johnson
Z (c)
Z (d)
Z (e)
Z
(f)
7e−x + 13
e−x
dx
3
2
1
+ 4− 7
x 5x
x
48 + u2
8u
2 + x2
dx
du
3 − x5 dx
3. Find f (x) using the following information.
f 0 (x) = 15x2 + 4x + 6, f (3) = 173.
3
Fall 2016,
©
Maya Johnson
4. The profit from the sale of a certain product is increasing at a rate given by
P 0 (x) = 390x1/3 , P (0) = 0
where x represents the number of weeks since the product was made available for sale. Determine
P (x).
4
Fall 2016,
©
Maya Johnson