INTEGRATION OF RATIONAL FUNCTIONS WITH SQUARE ROOTS

About "Integration of Rational Functions With Square Roots"

Integration of Rational Functions With Square Roots :

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

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

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

After having gone through the stuff given above, we hope that the students would have understood, "Integration of Rational Functions With Square Roots"

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