Page 33 - DMTH202_BASIC_MATHEMATICS_II

Basic Mathematics-II

Notes 4 2
x  x   1    2 1 2  
x
dx    x       dx
3
x  x 2   x x 2 x  1
1 dx
  xdx  2  dx   x  2 dx  2 
x x  1
x 2 1
  2log x   2log(x  1)
2 x
x 2 1 x
   2log
2 x x  1

Did u know? The integrand is an improper rational function. By “long division” of
polynomials, we can rephrase the integrand as the sum of a polynomial and a proper
rational function ”remainder”:

2
x  2
Example: Evaluate  3 dx
(x  1)(x  2)
Solution:

2
x
y
x
Let y      2
And dx  dy
2
(y  2)  2
  dy

(y  2 1)y 3
2
y  4y  6
  dy
y 5 (y  1)
2
y  4y  6 A B C D
How let 3   2  3 
y (y  1) y y y y  1
2

y  4y   Ay 2 (y  1) By (y  1) C (y  1) Dy 3

6

Putting y = 0
6 = C  C = 6
Again putting y = – 1
1 – 4 + 6 = – D  D = – 3
3
2
2
6


y  4y   A (y  y 2 ) B (y  y ) 6(y  1) 3y 3

Comparing the coefficients of y for A – 3 = 0 A = 3
A + B = 1 B = – 2
2
x  2   3 2 6 3  
Here  3 dx     2    dx
(x  1)(x  2)   (x  2) (x  2) (x  2) x  1  

28 LOVELY PROFESSIONAL UNIVERSITY