Page 21 - DMTH202_BASIC_MATHEMATICS_II

Basic Mathematics-II

Notes  Some people favor the substitution method over guess-and-check as it is more methodical,
but both methods attain the similar result.

1.4 Keywords

Anti-derivatives: The functions that could possibly have given function as a derivative are
called anti derivatives (or primitive) of the function.
Guess and Check method: 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.
Integration: 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.

1.5 Review Questions

1. Illustrate how integration can be defined as an Inverse Process of Differentiation.
2. Depict the concept of the Fundamental Theorem of Calculus with examples.

3. What is Guess-and-Check Method? Illustrate with examples.
b
w
4. Use substitution to state each of the following integrals as a multiple of a  (1/ )dw for
some a and b. Then evaluate the integrals.

1 x x /4 sinx
(a) 0  1 x 2 dw (b) 0  cosx dw

5. Find the antiderivatives of:
2
3
(a) sin(x x  1) (b) x 2 sin(x  1)

6. If appropriate, assess the integral x sin(x 2 )dx by substitution. If substitution is not
appropriate, mention it, and do not assess.

x 2
7. If suitable, assess the integral  2 dx by substitution. If substitution is not suitable,
1 x

mention it, and do not assess.
2
8. Find 4 (x x  1)dx by means of two methods:

(a) Do the multiplication first, and then anti-differentiate.
2
(b) Use the substitution w = x .
(c) Explain how the expressions from parts (a) and (b) are different. Are they both
correct?


9. Evaluate ( sin )x dx by using the method of substitution.
2
10. Evaluate sec x dx by using the method of substitution.


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