# HOW TO INTEGRATE A RATIONAL FUNCTION LINEAR IN THE NUMERATOR

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

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

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