# HOW TO INTEGRATE A RATIONAL FUNCTION LINEAR IN THE NUMERATOR

## About "How to Integrate a Rational Function Linear in the Numerator"

How to Integrate a Rational Function Linear in the Numerator :

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".

## Integration of Rational Functions Examples With Solutions

Question 1 :

Evaluate the following with respect to "x".

(3x + 1) / (2x2 - 2x + 3)

Solution :

∫ (3x + 1) / (2x2 - 2x + 3)dx

(3x + 1)   =   A(d/dx) (2x2 - 2x + 3) + B

3x + 1  =  A (4x - 2) + B  ----(1)

Equating the coefficients of x.

3  =  4A

A  =  3/4

Equating constant terms

1  =  -2A + B

1  =  -2(3/4) + B

1  =  -3/2 + B

B  =  1 + (3/2)  ===>  B  =  5/2

Applying the values of A and B in (1)

3x + 1  =  (3/4) (4x - 2) + (5/2)

By dividing each term by (2x2 - 2x + 3), we get

(3x + 1) / (2x2 - 2x + 3)dx

= (3/4)(4x-2)/(2x2-2x+3) dx+(5/2)1/(2x2-2x+3)dx

=  (3/4)log (2x2-2x+3)+(5/2)1/(2x2-2x+3)dx

2x2-2x+3 = 2[x- x + (3/2)]

=  2[x- 2x(1/2) + (1/2)2 - (1/2)2 + (3/2)]

=  2[(x-(1/2))2-(1/4) + (3/2)]

=  2 [(x-(1/2))+ ((5/2))2]

=  (3/4)log (2x2-2x+3)+(5/2)1/2 [((2x-1)/2)2+(5/2)2]dx

=  (3/4)log (2x2-2x+3)+(5/2)tan-1[(2x-1)/5]

Question 2 :

Evaluate the following with respect to "x".

(2x + 1) / √(9 + 4x - x2)

Solution :

∫(2x + 1) / √(9 + 4x - x2dx

(2x + 1)   =   A(d/dx) (9 + 4x - x2) + B

2x + 1  =  A (-2x + 4) + B  ----(1)

Equating the coefficients of x.

2  =  -2A

A  =  -1

Equating constant terms

1  =  4A + B

1  =  4(-1) + B

1  =  -4 + B

B  =  1 + 4 ===>  B  =  5

Applying the values of A and B in (1)

2x + 1  =  -1 (-2x + 4) + 5

By dividing each term by (2x2 - 2x + 3), we get

(2x + 1) / √(9 + 4x - x2dx

= -1(-2x+4)/√(9 + 4x - x2) dx + 5 1/√(9 + 4x - x2)dx

=  -2 log √(9 + 4x - x2) + 5 1/√(9 + 4x - x2)dx

√(9 + 4x - x2) = -[x- 4x - 9]

=  [x- 2x(2) + 22 - 22 - 9]

=  [(x - 2)- 13]

=  [(x - 2)- (13)2]

=  -2 log √(9 + 4x - x2) + 5 1/ [(x - 2)- (13)2]dx

=  -2 log √(9 + 4x - x2) + 5 sin-1 [(x - 2)/13]

=  + 5 sin-1 [(x - 2)/13] - 2 log √(9 + 4x - x2) + c After having gone through the stuff given above, we hope that the students would have understood, "How to Integrate a Rational Function Linear in the Numerator"

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