MODULE 1:
GENERAL ANTIDIFFERENTIATION
FORMULAS
1.1 BASIC INTEGRATION
FORMULAS
Definition of Integral Calculus
A branch of Mathematics concerned with the
theory and applications (as in the determination of
lengths, areas and volumes and in the solutions of
differential equations) of integrals and integration.
(Merriam-Webster)
- also known as anti-differentiation.
- the process of finding a function whose derivative is
known.
- inverse process of differentiation.
The anti-derivative or the indefinite integral is defined as
The following illustrate further the integral concept:
Basic Integration Formulas
1.
Examples:
a. dx =
b.  dy =
2.  a dx = a dx = ax + C
Examples:
a. 2dx =
b.
c.
3.
Examples
a.
b.
c.
4.
Examples:
a.
b.
c.
1.2 GENERAL POWER RULE
U-Substution
The method of u-substitution is a method for
algebraically simplifying the form of a function so that its
antiderivative can be easily recognized.
Examples:
a.
b.
c.
Examples (U – Substitution)
1.
2.
3.
4.
1.3 POWER RULE WITH NEGATIVE
ONE EXPONENT
Power Formula with Negative One Exponent
The Power Formula with negative one exponent is given
by
Examples:
1.
2.
3.
4.
DISCUSSION 1.1
1.4 Definition of Definite Integral
Definite Integral
Let f be a function on a closed interval [a,b], then
the definite integral of f from a to b is denoted by:
Examples:
1.
2.
3.
4.
Using theorem 3
1.5 Odd and Even Functions
Even Functions
A function that remains unchanged when x is replaced
by -x is called an even functions.
f(−x)=f(x)
Geometrically, the curve of an even function is
symmetric with respect to the y-axis.
Examples:
Odd Functions
A function such that
f(−x)=−f(x)
is called an odd function. Geometrically , the curve of an
odd function is symmetric with respect to the origin.
Examples
THEOREMS
1. If f(x) is an even function, then
Example:
Using the theorem
2. If f(x) is an odd function, then
Example:
1.6 Change of Limits
Change of Limits
When evaluating an integral using u-substitution, an
expression involving the original variable is replaced by a
new variable. The end result is a simpler integral.
When we use u-substitution to evaluate a definite integral,
we must change each part of the integral to use the new
variable (usually u) instead of the old variable (often x).
There are three pieces that must be changed:
•The
function
itself
(change
an
expression
involving x into u)
•The differential (change dx into an expression
involving du)
•The bounds (change the bounds of integration from
values for x into values for u
Examples:
1.
2.
3.
NEXT:
MODULE 2