Page 75 - DMTH202_BASIC_MATHEMATICS_II

Basic Mathematics-II

Notes So, we can say, the Substitution Rule for Definite Integrals state: If f is continuous on the range
of u = g(x) and g’(x) is continuous on [a, b], then
b
b g ( )
x
g
x
u
g
 f ( ( )) ´( )dx   f ( )du
a g ( ) a
Did u know? One of the methods of performing the assessment is perhaps the most
understandable at this point, but also has a point in the procedure where we can dig up in
problem if we aren’t paying awareness.

Self Assessment

Fill in the blanks:
1. The first step in performing a definite integral is to calculate the ....................... integral.
2. The steps for performing integration by substitution for definite integrals are .......................
as the steps for integration by substitution for indefinite integrals.
3. If f is continuous on the range of u = g(x) and g’(x) is continuous on [a, b], then .......................
State whether the following statements are true or false:

4. Performing integration by substitution for definite integrals is different from performing
integration by substitution for indefinite integrals.
5. The new bounds of integration are located by plugging in the lower bound.

5.2 Use Substitution to Find Definite Integrals

To Use Substitution to find Definite Integrals, you are required to perform either:
 Calculate the indefinite integral, articulating an antiderivative in terms of the original
variable, and then assess the consequence at the original limits, or

 Translate the original limits to new limits in provisions of the new variable and do not
translate the antiderivative back to the original variable.
Let us now describe both methods of performing the evaluation step.

Consider the following definite integral.
We will illustrate here, the evaluation of the given integral by means of two different
methods.

0 2 3
2  2t 1 4t dt


Let’s begin off from first method of dealing with the assessment step.
!

Caution We are required to be cautious with this method as there is a point in the procedure
where if we aren’t paying awareness we’ll obtain the wrong solution.

70 LOVELY PROFESSIONAL UNIVERSITY