Question

Evaluate the definite integral:

`int_1^2 (5x^2)/(x^2 + 4x + 3)`

Answer 1

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)`