1. In this handout, 4. 7 Antiderivatives 5
In this handout, 4.7 Antiderivatives 5.3 Evaluating Definite Integrals 5.4 The Fundamental Theorem of Calculus

2. Antiderivatives
Definition: A function F is called an antiderivative of f on an interval I if F’(x) = f(x) for all x in I.
Example: Let f(x)=x3. If F(x) =1/4 * x4 then F’(x) = f(x)
Theorem: If F is an antiderivative of f
on an interval I, then the most general
antiderivative of f on I is
F(x) + C
where C is an arbitrary constant.
F is an antiderivative of f

3. Table of particular antiderivatives
Function
Antiderivative
xr, r  -1
xr+1/(r+1)
1/x
ln |x|
sin x
- cos x ex
cos x ax
ax / ln a
sec2 x
tan x
tan-1 x
sin-1 x
af + bg
aF + bG
In the last and highlighted formula above we assume that F’ = f and G’ = g. The coefficients a and b are numbers.
These rules give particular antiderivatives of the listed functions. A general antiderivative can be obtained by adding a constant .

4. Computing Antiderivatives
Problem
Solution
Use here the formula (aF + bG)’ = aF’ + bG’

5. Analyzing the motion of an object using antiderivatives
Problem
A particle is moving with the given data. Find the position of the particle.
a(t) = t -3t2, v(0) = 2, s(0) = 5
v(t) is the antiderivative of a(t): v’(t) = a(t) = t -3t2
Antidifferentiation gives v(t) = 10t + 1.5t2 – t3 + C
v(0) = 2 implies that C=2; thus, v(t) = 10t + 1.5t2 – t3 + 2
s(t) is the antiderivative of v(t): s’(t) = v(t) = 10t + 1.5t2 – t3 + 2
Antidifferentiation gives s(t) = 5t t3 – 0.25t4 + 2t + D
s(0) = 5 implies that D=5; thus, s(t) = 5t t3 – 0.25t4 + 2t + 5
Solution

6. Evaluation Theorem
Theorem: If f is continuous on the interval [a, b], then
where F is any antiderivative of f, that is, F’=f.
Example:
since F(x)=1/3 * x3 is an antiderivative of f(x)=x2

7. Indefinite Integral
Indefinite integral is a traditional notation for antiderivatives
Note: Distinguish carefully between definite and indefinite integrals. A definite integral is a number, whereas an indefinite integral is a function (or family of functions). The connection between them is given by the Evaluation Theorem:

8. Some indefinite integrals

9. The Fundamental Theorem of Calculus
The theorem establishes a connection between the two branches of calculus: differential calculus and integral calculus:
Theorem: Suppose f is continuous on [a, b].
Example of Part 1: The derivative of is

10. Average value of a function
The average value of function f on the interval [a, b] is defined as
Note: For a positive function, we can think of this definition as saying area/width = average height
Example: Find the average value of f(x)=x3 on [0,2].

11. The Mean Value Theorem for Integrals
Theorem: If f is continuous on [a, b], then there exists a number c in [a, b] such that
Example: Find c such that fave=f(c) for f(x)=x3 on [0,2].
From previous slide, f(c)=fave=2.
Thus, c3=2, so