INTEGRATION OF RATIONAL FUNCTIONS WITH QUADRATIC DENOMINATOR

About "Integration of Rational Functions with Quadratic Denominator"

Integration of Rational Functions with Quadratic Denominator :

Here we are going to see some example problems to understand how to evaluate integration of rational functions with quadratic denominator.

To know the formulas used in integration, please visit the page "Integration Formulas for Class 12".

∫ dx/(a2-x2)  =  (1/2a) log [(a + x)/(a - x)] + c

∫ dx/(x2-a2)  =  (1/2a) log [(x - a)/(x + a)] + c

∫ dx/(a2+ x2)  =  (1/a) tan-1 (x/a) + c

∫ dx/(a2- x2)  =  sin-1 (x/a) + c

∫ dx/(√x2- a2)  =  log (x + (√x2- a2)) + c

∫ dx/(√x2+ a2)  =  log (x + (√x2+ a2)) + c

∫ √(a2- x2)  =  (x/2) √(a2- x2) + (a2/2) sin-1(x/a) + c

∫ √(x2- a2)  =  (x/2) √(x2- a2) - (a2/2) log(x+√(x2- a2))+c

∫ √(x2+a2)  =  (x/2) √(x2+ a2) + (a2/2) log(x+√(x2+ a2))+c

Integration of Rational Functions with Quadratic Denominator - Examples

Question 1 :

Integrate the following with respect to x:

1/(4 - x2)

Solution :

  =  ∫1/(4 - x2) dx

=  ∫1/(22 - x2) dx 

1/(a2 - x2) dx   =   (1/2a) log [(a + x)/(a - x)] + c

=  (1/2(2)) log [(2 + x) / (2 -  x)] + c

=  (1/4) log [(2 + x) / (2 - x)] + c

Question 2 :

Integrate the following with respect to x:

1/(25 - 4x2)

Solution :

  =  ∫1/(25 - 4x2) dx

  =  ∫1/(52 - (2x)2) dx

here a = 5 and x = 2x

  =  (1/22(5))log [(5 + 2x) / (5 - 2x)] + c

  =  (1/20)log [(5 + 2x) / (5 - 2x)] + c

Question 3 :

Integrate the following with respect to x:

1/(9x2 - 4)

Solution :

  =  ∫ 1/((3x)2 - 22) dx

∫ dx/(x2-a2)  =  (1/2a) log [(x - a)/(x + a)] + c

Here x = 3x and a = 2

  =  (1/2⋅3(2))log [(3x - 2) / (3x - 2)] + c

  =  (1/12)log [(3x - 2) / (3x - 2)] + c

Question 4 :

Integrate the following with respect to x:

1/(6x - 7 - x2)

Solution :

  =  ∫ 1/(6x - 7 - x2)dx

  =  -∫ 1/(x2 - 6x + 7)dx

x2 - 6x + 7  =  x2 - 2x(3) + 32 - 32+ 7

=  (x - 3)2 - 9 + 7

=  (x - 3)2 - 2

x2 - 6x + 7 =  (x - 3)2 - (2)2

 -∫ 1/(x2 - 6x + 7)dx  =   -∫ 1/[(x - 3)2 - (2)2]dx

 ∫1/[22 (x - 3)2]dx

1/(a2 - x2) dx   =   (1/2a) log [(a + x)/(a - x)] + c

  =  (1/22) log [(2 + (x - 3)) / (2 - (x - 3))]

  =  (1/22) log [(2 + x - 3) / (2 - x + 3)] + c

  =  (1/22) log [(2 - 3 x) / (2 + 3 x)] + c

After having gone through the stuff given above, we hope that the students would have understood, "Integration of Rational Functions with Quadratic Denominator"

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