INTEGRATION OF LINEAR IN NUMERATOR AND QUADRATIC IN THE DENOMINATOR

About "Integration of Linear in Numerator and Quadratic in the Denominator"

Integration of Linear in Numerator and Quadratic in the Denominator :

Here we are going to see some example problems to understand evaluating integration of linear in the numerator and quadratic in the denominator.

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

Integrating Rational Functions Linear in the Numerator and Quadratic in the Denominator - Examples

Question 1 :

Evaluate the following with respect to "x".

(2x - 3) / (x² + 4x - 12)

Solution :

 ∫(2x - 3) / (x² + 4x - 12) dx

(2x - 3)   =   A(d/dx) (x² + 4x - 12) + 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  =  -7

Applying the values of A and B in (1)

2x - 3  =  1 (2x + 4) - 7

By dividing each term by , we get

∫(2x-3)/(x²+4x-12) dx 

  = ∫(2x-4)/(x²+4x-12) dx - 7 ∫1/(x²+4x-12) dx

  =  log (x²+4x-12) - 7∫1/(x²+4x-12) dx

x²+4x-12  =  x² + 2x(2) + 2² - 2² -12

 =  (x + 2)² + 4 - 12

 =  (x + 2)² - 8

 =  (x + 2)² - 4²

  =  log (x²+4x-12) - 7∫1/[(x + 2)² - 4²] dx

  =  log (x²+4x-12) - 7 [1/2(4) log(x + 2 - 4) / (x + 2 + 4)]

  =  log (x²+4x-12) - (7/8) [log(x-2) / (x + 6)] + c

Question 2 :

Evaluate the following with respect to "x".

(5x - 2) / (2 + 2x + x²)

Solution :

 ∫(5x - 2) / (2 + 2x + x²) dx

(5x - 2)   =   A(d/dx) (2 + 2x + x²) + B

5x - 2  =  A (2 + 2x) + B  ----(1)

Equating the coefficients of x.

5  =  2A

A  =  5/2

Equating constant terms

-2  =  2A + B

-2  =  2(5/2) + B

-2  =  5 + B

B  =  -2 - 5  ===>  B  =  -7

Applying the values of A and B in (1)

5x - 2  =  (5/2) (2 + 2x) - 7

By dividing each term by , we get

∫(5x - 2) / (2 + 2x + x²) dx 

  =  (5/2) ∫(2x + 2) / (2 + 2x + x²) - 7 ∫1/(2 + 2x + x²) dx

  =  log (2 + 2x + x²) - 7∫1/(2 + 2x + x²) dx

2 + 2x + x²  =  x² + 2x(1) + 1² - 1² + 2

 =  (x + 1)² - 1 + 2

 =  (x + 1)² + 1

  =  (5/2)log (2 + 2x + x²) - 7∫1/[(x + 1)² + 1] dx

  =  (5/2) log (2 + 2x + x²) - 7 tan^-1 (x + 1) + c

  =  (5/2) log (2 + 2x + x²)  - 7 tan^-1 (x + 1) + c

After having gone through the stuff given above, we hope that the students would have understood, "Integration of Linear in Numerator and Quadratic in the Denominator"

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