INTEGRATION OF  RATIONAL FUNCTION WITH SQUARE ROOT IN DENOMINATOR

About "Integration of Rational Function With Square Root in Denominator"

Integration of Rational Function With Square Root in Denominator  :

Here we are going to see some example problems to understand integration of rational function with square root in denominator

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

Integration of Rational Functions Examples With Solutions

Question 1 :

Evaluate the following with respect to "x".

(x + 2) / √(x2 - 1)

Solution :

(x + 2) / √(x2 - 1)  =  (x/√(x2 - 1)) dx + 2∫(1/√(x2 - 1)) dx

(x/√(x2 - 1)) dx 

x2 - 1  =  t

2x dx  =  dt

x dx  =  dt/2

(x/√(x2 - 1)) dx  =  ∫(1/√t) dt

  =  t-1/2 (dt/2)

  =  (1/2) t1/2/(1/2)

  =  (x2 - 1)   ---(1)

2∫(1/√(x2 - 1)) dx  =  2 log(x + √(x2 - 1))   ------(2)

(1) + (2)

  =  (x2 - 1) + 2 log(x + √(x2 - 1)) + c

Question 2 :

Evaluate the following with respect to "x".

(2x + 3) / √(x2 + 4x + 1)

Solution :

 ∫(2x + 3) / √(x+ 4x + 1) dx

(2x + 3)   =   A(d/dx) (x+ 4x + 1) + B

2x + 3  =  A (2x + 4) + B  ----(1)

Equating the coefficients of x.

2  =  2A

A  =  1

Equating constant terms

3  =  4A + B

3  =  4(1) + B

3  =  4 + B

B  =  3 - 4 ===>  B  =  -1

Applying the values of A and B in (1)

2x + 3  =  1 (2x + 4) - 1

By dividing each term by √(x+ 4x + 1), we get

(2x + 1) / √(x+ 4x + 1) dx

  =  (2x+4)/√(x+ 4x + 1) dx - 1 1/√(x+ 4x + 1)dx

  =  2 log (x+ 4x + 1) - 1 1/(x+ 4x + 1)dx

(x+ 4x + 1)  =   [x+ 2x(2) + 22 - 22 + 1]

  =  [(x + 2)- 3]

  =  [(x + 2)32]

  =  2 log (x+ 4x + 1) - 1 1/[(x + 2)32] dx

  =  2 log (x+ 4x + 1) - log (x + 2) + (x+ 4x + 1) + c

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