Page 83 - DMTH202_BASIC_MATHEMATICS_II

Basic Mathematics-II

Notes 9. There is a point in the procedure where if we aren’t paying awareness we’ll obtain the
............................... solution.
10. The second solution method isn’t actually all that dissimilar from the ...............................
method.
11. Getting back to t’s before performing the substitution is the standard step in the
............................... procedure, but it is frequently forgotten when performing definite integrals.
12. If we don’t move back to t’s we will have a little problem in that one of the evaluations
will finish up providing us a ............................... .
13. Be cautious with definite integrals and be on the lookout for ............................... problems.
State whether the following statements are true or false:
14. In first method, when performing a substitution we want to remove all the t’s in the
integral and write all in terms of u.
15. We are required to be alert with the first method as there is a point in the process where if
we aren’t paying awareness we’ll obtain the wrong solution.

5.3 Summary

 The first step in performing a definite integral is to calculate the indefinite integral and
that hasn’t tainted.

 The steps for performing integration by substitution for definite integrals are the similar
as the steps for integration by substitution for indefinite integrals apart from we must
alter the bounds of integration and we do not require subbing back in for u.

 The Substitution Rule for Definite Integrals state: If f is continuous on the range of u = g(x)
b g   b
and g’(x) is continuous on [a, b], then   f g x g ´  f   u du
    x dx 
a g   a
 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.
 Also, another method is to translate the original limits to new limits in provisions of the
new variable and do not translate the antiderivative back to the original variable.

 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.
 The second solution method isn’t actually all that dissimilar from the first method.
 Both are valid solution methods and each include their uses.

5.4 Keyword

Substitution Rule: The Substitution Rule for Definite Integrals state: If f is continuous on the
b g   b
    x dx 
range of u = g(x) and g’(x) is continuous on [a, b], then   f g x g ´  f   u du
a g   a
5.5 Review Questions

1. Illustrate the steps for performing integration by substitution for definite integrals with
example.

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