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) t^1/2/(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) / √(x² + 4x + 1) dx

(2x + 3)   =   A(d/dx) (x² + 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 √(x² + 4x + 1), we get

∫(2x + 1) / √(x² + 4x + 1) dx

  =  ∫(2x+4)/√(x² + 4x + 1) dx - 1 ∫1/√(x² + 4x + 1)dx

  =  2 log √(x² + 4x + 1) - 1 ∫1/√(x² + 4x + 1)dx

√(x² + 4x + 1)  =   [x² + 2x(2) + 2² - 2² + 1]

  =  [(x + 2)² - 3]

  =  [(x + 2)² - √3²]

  =  2 log √(x² + 4x + 1) - 1 ∫1/√[(x + 2)² - √3²] dx

  =  2 log √(x² + 4x + 1) - log (x + 2) + √(x² + 4x + 1) + c

Related topics

• Indefinite Integral Using Properties
• Integration of Rational Functions with Quadratic Denominator
• Integration of Rational Functions With Square Roots
• Integration of Linear in Numerator and Quadratic in the Denominator
• How to Integrate a Rational Function Linear in the Numerator
• Integration of Rational Function With Square Root in Denominator

After having gone through the stuff given above, we hope that the students would have understood, "Integration of Rational Function With Square Root in Denominator"

Apart from the stuff given in "Integration of Rational Function With Square Root in Denominator", if you need any other stuff in math, please use our google custom search here.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.