Introduction to Anti-differentiation
DEMPC Retooling/Refresher
Course 2024
1
Introduction to Anti-differentiation
To find a function F whose derivative
is f(x) = 3x2, you might use your
knowledge of derivatives to conclude
that
The function F is an antiderivative of f .
DEMPC Retooling/Refresher
Course 2024
2
Introduction to Anti-differentiation
Note that F is called an
antiderivative of f, rather than the
antiderivative of f. To see why,
observe that
are all derivatives of f (x) = 3x2. In
fact, for any constant C, the
function F(x)= x3 + C is an
antiderivative of f.
DEMPC Retooling/Refresher
Course 2025
3
Intro to Anti-differentiation
Using Theorem 4.1, you can represent the entire family of
antiderivatives of a function by adding a constant to a known
antiderivative.
For example, knowing that Dx [x2] = 2x, you can represent the
family of all antiderivatives of f(x) = 2x by
G(x) = x2 + C
where C is a constant. The constant C is called the constant
of integration.
DEMPC Retooling/Refresher
Course 2024
4
Definite and Indefinite Integrals
When solving a differential equation of the form
it is convenient to write it in the equivalent differential form
The operation of finding all solutions of this equation is
called antidifferentiation (or indefinite integration) and is
denoted by an integral sign ∫.
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Course 2024
5
Indefinite Integrals
The general solution is denoted by
The expression ∫f(x)dx is read as the antiderivative of
f with respect to x. So, the differential dx serves to
identify x as the variable of integration. The term
indefinite integral is a synonym for antiderivative.
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Course 2025
6
Definite Integral
Consider two functions f and g that are continuous
on the interval [a, b].
Also, the graphs of both f and g
lie above the x-axis, and the graph
of g lies below the graph of f,
as shown in the Figure.
The Definite Integral essentially solves the area
between the 2 curves.
Best Practice during classes is to slowly introduce
this concept and define the limits of integration as
boundaries and the procedure would be the same
with the indefinite integral except for C.
DEMPC Retooling/Refresher
Course 2025
7
Basic Integration Rules
DEMPC Retooling/Refresher
Course 2025
8