INTEGRATION OF RATIONAL FUNCTIONS WITH SQUARE ROOTS

Integrals With Square Roots in Denominator - Examples

Question 1 :

Integrate the following with respect to x:

1/[(x + 1)² - 25]

Solution :

  =  ∫1/[(x + 1)² - 25] dx

=  ∫1/[(x + 1)² - 5²] dx 

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

Here x = x + 1 and a = 5

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

=  (1/10) log [(x - 4) / (x + 6)] + c

Question 2 :

Integrate the following with respect to x:

1/√(x² + 4x + 2)

Solution :

  =  ∫1/√(x² + 4x + 2) dx

x² + 4x + 2  =  x² + 2x(2) + 2² - 2² + 2

=  (x + 2)² - 4 + 2

x² + 4x + 2  =  (x + 2)² - 2

=  (x + 2)² - √2²

1/√[(x + 2)² - √2²]

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

Here x = x + 2 and a = √2

  =  log (x + 2 + √[(x + 2)² - √2²) + c.

  =  log (x + 2 + √(x² + 4x + 2) + c

Question 3 :

Integrate the following with respect to x:

1/√((2+x)² - 1)

Solution :

  =  ∫ 1/√((2+x)² - 1) dx

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

Here x = 2 + x and a = 1

  =  log (2 + x + √[(2 + x)² - 1²) + c.

  =  log (x + 2 + √( (x + 2)² - 1 ) + c

Question 4 :

Integrate the following with respect to x:

1/√(x² - 4x + 5)

Solution :

  =  ∫ 1/√(x² - 4x + 5) dx

x² - 4x + 5  =  x² - 2x(2) + 2² - 2² + 5

=  (x - 2)² - 4 + 5

x² - 4x + 5  =  (x - 2)² + 1

=  (x - 2)² + 1

1/√(x - 2)² + 1

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

Here x = x - 2 and a = 1

  =  log (x - 2 + √[(x - 2)² - 1²) + c.

  =  log (x - 2 + √(x² - 4x + 5)) + c

Question 5 :

Integrate the following with respect to x:

1/√(9 + 8x - x²)

Solution :

  =  ∫ 1/√(9 + 8x - x²)

 dx

9 + 8x - x²  =  - (x² - 8x - 9)

=  -(x² - 2x(4) + 4² - 4² - 9)

=  - [(x - 4)² - 16 - 9]

9 + 8x - x² =  -[(x - 4)² - 25]

=   5²- (x - 4)²

1/√[ 5²- (x - 4)²]

∫ dx/(√a²- x²)  =  sin^-1(x/a) + c

Here x = x - 4 and a = 5

  =  sin^-1 [(x - 4)/5] + c

  =  sin^-1 [(x - 4)/5] + c

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