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Mathematics class XII

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Unit –III (Calculus)

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Chapter – Indefinite Integral

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Sub Topic – Integration by Partial

Fractions

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Outline

  • Partial Fraction of the Type when degree of the numerator less then

or equal to the degree of the denominator and

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  • Denominator can be expressed as product of distinct linear factor

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  • Denominator can be expressed as product of linear factor in which some

Or all are repeating

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  • Denominator is not completely reducible to linear factor

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  • when degree of the numerator is greater than or equal to the degree of

the denominator and

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  • Examples

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  • Assignments

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INTEGRATION BASED ON PARTIAL FRACTION

METHOD OF RESOLVING A GIVEN RATIONAL FUNCTION INTO PARTIAL FRACTION

TYPE I

This method is applicable when

  • The degree of the numerator is less than the degree of the denominator
  • The denominator

can be expressed as

the product of distinct linear factors.

Suppose

= (x + A)(x + B)(x + C)....... (x + k), then

simplifying and comparing

coefficients of x,

...., constant, etc, we can determine the value of

,

....

.

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TYPE II

This method is applicable when

  • The degree of the numerator is less than the degree of the denominator
  • The denominator

can be expressed as the product of linear factors in

which one or some are repeating.

Suppose

= (x + A)

,

simplifying and comparing coefficients of x,

, constant, etc, we can determine the value of

,

and

.

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TYPE III :

This method is applicable when

  • The degree of the numerator is less than the degree of the denominator
  • The denominator

cannot be expressed as the product of

distinct linear factors

Then we write

simplifying and comparing coefficients of x,

, etc, we can determine the value of

,

and

.

, constant

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TYPE IV

This method is applicable when

  • The degree of the numerator is greater than the degree

of the denominator

  • Divide the numerator with denominator till the degree of

the numerator become less than the degree of denominator and

than apply any one of the three above methods depending on the

type of denominator.

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Assignment