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".
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"
Apart from the stuff given in "Integration of Linear in Numerator and Quadratic in the 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.
©All rights reserved. onlinemath4all.com
May 30, 23 11:19 AM
May 30, 23 10:38 AM
May 26, 23 12:27 PM