Page 9 - DMTH202_BASIC_MATHEMATICS_II

Basic Mathematics-II

Notes 3. ..................... is defined as the inverse process of differentiation.
4. The Fundamental Theorem of Calculus identifies the ..................... among the processes of
differentiation and integration.

5. The + C occurs since the derivative of any constant term is .....................
6. ..................... is known as the constant of integration.
7. The value of C can be instituted when suitable additional information is given, and this
provides a specific ..................... .
8. The ..................... relationship among differentiation and integration means that, for each
statement regarding differentiation, we can write down an equivalent statement
concerning integration.
d
x

9. (sin )  cosx , so cosxdx =..................... .
dx
1.2 Integration by Substitution

Integration can be simply performed by means of integration formulas, If the integral is in the
typical form where we can simply pertain formulas. But if the function which is to be integrated
is not in the typical form then it is either tough or impracticable to utilize integration formulas
to integrate. In that case we are required to use Integration by Substitution method to integrate
a specified function.
In the process of Integration by substitution, 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.

1.2.1 The Guess-and-Check Method

The Guess-and-Check 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.

Did u know? If we get the predictable result, then we’re completed; or else, we rework the
guess and check again.
The method of guess-and-check is functional in reversing the chain rule. The chain rule shows:
inside
d 
f'
(f(g(x)))   (g(x)). g'(x)

dx
Derivative of outside Derivative of inside
Therefore, any function which is the consequence of applying the chain rule is the product of two
factors: the “derivative of the outside” and the “derivative of the inside.” If a function contains
this form, its anti derivative is f (g(x)).

2
3
Example: Find 3x cos(x )dx.

Solution:
3
2
The function 3x cos(x ) appears as the result of applying the chain rule: there is an “inside”
2
3
function x and its derivative 3x occurs as a factor. As the outside function is a cosine which has
3
a sine (x ) as an ant derivative, we guess for the anti derivative. Differentiating to check provides.
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