Page 20 - DMTH202_BASIC_MATHEMATICS_II

Unit 1: Integration

Observe that the substitution in the previous example again converts the inside of the messiest Notes
function into something easy.

!

Caution As the derivative of the inside function is not waiting for us, we have to solve for
x so that we can obtain dx completely in terms of w and dw.

Self Assessment

Fill in the blanks:

10. In the process of Integration by.............................., we reduce an integral in non-standard
form into a integral in standard form by altering the variable into a new variable with
appropriate substitution.
11. The .............................. Method, a high-quality strategy for locating simple ant derivatives
is to guess an answer (by means of knowledge of differentiation rules) and then check the
answer by means of differentiating it.
12. The method of guess-and-check is functional in .............................. the chain rule.
13. Any function which is the consequence of applying the chain rule is the ..............................
of two factors: the “derivative of the outside” and the “derivative of the inside.”
14. .............................. a function frequently includes “correcting for” constant factors.
15. We can apply the substitution method when a constant factor is absent from the derivative
of the .............................. function.

1.3 Summary

 Integral Calculus is motivated by the problem of defining and calculating the area of the
region bounded by the graph of the functions.
 The functions that could possibly have given function as a derivative are called anti
derivatives (or primitive) of the function.

 The formula that gives all the anti derivatives is called the indefinite integral of the
function and such process of finding anti derivatives is called integration.

 There is a connection, known as the Fundamental Theorem of Calculus, between indefinite
integral and definite integral which makes the definite integral as a practical tool for
science and engineering.

 The Fundamental Theorem of Calculus defines the relationship among the processes of
differentiation and integration.

 The inverse relationship among differentiation and integration means that, for each
statement about differentiation, we can write down a matching statement regarding
integration.

 In the method of Integration by substitution, we reduce a integral in non-standard form
into a integral in standard form by altering the variable into a new variable with appropriate
substitution.
 The Guess-and-Check Method is a high-quality strategy for locating simple ant derivatives
is to guess an answer (by means of knowledge of differentiation rules) and then check the
answer by differentiating it.

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