INTEGRATION OF RATIONAL FUNCTIONS WITH QUADRATIC DENOMINATOR

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

  =  (1/2⋅2(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

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

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

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

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.