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Question
Evaluate the definite integral:
`int_1^2 (5x^2)/(x^2 + 4x + 3)`
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Solution
Since the degree of numerator and denominator are same
∵ The fraction is improper. To make it proper, we have to divide 5x2 by x2 + 4x + 3.
`x^2+4x+3overline(")"5x^2 "(")5`
`" "5x2+20x+15`
`" "underline(-" "-" "- )`
`" "underline(- 20x - 15 )`
∴ `I = int_1^2 (5 + (-20 + 15)/ (x^2 + 4x + 3)) dx`
Let `(20x + 15)/(x^2 + 4x + 3)`
`= (20x + 15)/((x + 1)(x + 3)) ≡ A/(x+1) + B/(x + 3)`
⇒ 20x + 15 ≡ A (x + 3) + B (x + 1) ....(i)
Putting x = -1 in (i), we get
-20 + 15 = A (-1 + 3)
⇒ -5 = 2A
⇒ `A = (-5)/2`
Putting x = -3 in (i), we get
-60 + 15 = B (- 3+ 1)
⇒ -45 = - 2 B
⇒ `B = 45/2`
∴ `I = int_1^2 (5 + 5/ (2 (x + 1)) - 45/(2(x + 3))) dx`
`= [5x + 5/2 log (x + 1) - 45/2 log (x + 3)]_1^2`
`= 5 (2 - 1) + 5/2 [log 3 - log 2] - 45/2 [log 5 - log 4]`
`= 5 + 5/2 log 3/2 - 45/2 log 5/4`
`= 5 - 5/2 (9 log 5/4 - log 3/2)`
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