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Integral Calculus: INTRODUCTION: Just As Subtraction Is The Inverse of

The document discusses integral calculus, which is the inverse operation of differentiation. It defines antiderivatives as functions whose derivatives are a given function, and indefinite integrals as finding antiderivatives using properties like: the integral of a derivative is the original function plus a constant; a constant can be moved outside the integral sign but not a variable. Examples demonstrate properties like the power rule and substitution method. Integrals leading to natural logarithmic functions and exponential functions are also covered, with examples of applying relevant formulas.

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Charlstom Moreno
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139 views10 pages

Integral Calculus: INTRODUCTION: Just As Subtraction Is The Inverse of

The document discusses integral calculus, which is the inverse operation of differentiation. It defines antiderivatives as functions whose derivatives are a given function, and indefinite integrals as finding antiderivatives using properties like: the integral of a derivative is the original function plus a constant; a constant can be moved outside the integral sign but not a variable. Examples demonstrate properties like the power rule and substitution method. Integrals leading to natural logarithmic functions and exponential functions are also covered, with examples of applying relevant formulas.

Uploaded by

Charlstom Moreno
Copyright
© Attribution Non-Commercial (BY-NC)
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as DOCX, PDF, TXT or read online on Scribd
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INTEGRAL CALCULUS

INTRODUCTION: Just as subtraction is the inverse of addition, division is the inverse operation of multiplication, extracting roots is the inverse operation of raising to powers, the process called antidifferentiation or integration is the inverse operation of differentiation. OBJECTIVE: To be able to know the definition of antiderivative and apply this definition in finding the function whose derivative is given by using the fundamental properties of indefinite integrals. ANTIDIFFERENTIATION- the process of finding a function whose derivative is known. The required function is called an anti-derivative or an integral of the given function which is called the integrand. If F(x) is a function whose derivative is f(x) the relation between the two is expressed by writing ( ) ( )

where the symbol: is the integral sign

f(x) is the integrand F(x) is the particular integral C constant of integration

F(x) + C indefinite integral of f(x)

The integral sign indicates that we are to perform the operation of integration on f(x) dx, that is we are to find a function whose differential is f(x) dx. A. GENERAL PROPERTIES OF INDEFINITE INTEGRALS 1. The integral of the differential of a function u is u plus an arbitrary constant C. 2. A constant may be written before integral sign but not a variable function.

3. Power formula: If n is not equal to minus one, the integral of du is obtained by adding one to the exponent and dividing by the new component. 4. The integral of the sum of several functions is equal to the sum of the integrals of separate functions. ) = ( EXAMPLES: Perform the anti-differentiation and check by finding the derivative in your answer.

1. ; d(x + C) = dx; since differential of C is zero 2. Solution: we may take 5 outside the sign of integration since it is a constant. , ( )

We may not take x outside the integration sign; because x is a variable. 3. Solution: by applying the 2nd and 3rd properties. , since

Power formula can be applied in d(x) = dx and n = 3

( ( )

Solution: applying 4th property: ( )

SUBSTITUTION METHOD: CHAIN RULE Quite often, the process of integration can be simplified by use of a substitution or change of variable. The purpose of substituting a new variable is to bring the problem to a form for which the standard formula,

+c,n

can be applied.

EXAMPLES:

1. ( 2.

) (2x +3)dx
( )

3. 4.

5. ( 6. ( 7. 8 (

) ) )

9. 10.

(
( ) )

INTEGRALS LEADING TO NATURAL LOGARITHMIC FUNCTIONS

= ln| | + c

Take note that this formula can be applied if the numerator is exactly the differential of the denominator.

EXAMPLES:

1. 2.

3. 4. 5. 6. 7.

( ( )(

) ) )

8.

9.

10.

INTEGRALS INVOLVING EXPONENTIAL FUNCTIONS

+c

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