Evaluate: `int(x^2)/(x^4-x^2-12)\ dx`

Answer

N/a

Explanation

Let `I = int(x^(2))/(x^(4)-x^(2)-12)dx`
`= int(x^(2))/(x^(4)-4x^(2)+3x^(2)-12)dx`
`= int(x^(2)dx)/(x^(2)(x^(2)-4)+3(x^(2)-4))`
`= int(x^(2)dx)/((x^(2)-4)(x^(2)+3))`
Now, `(x^(2))/((x^(2)-4)(x^(2)+3)), ["let" x^(2)= t]`
`rArr (t)/((t-4)(t+3)) = (A)/(1-4) + (B)/(t+3)`
`rArr t = A (t+3) + B(t-4)`
Om comparing the coefficient of t on both sides, we get
`A + B = 1 "...."(i)`
`rArr 3A + 4B = 0 "....."(ii)`
` rArr 3 (1+B) - 4 B = 0`
`rArr 3 - 3B - 4B = 0`
`rArr 7B = 3`
`rArr B = 3/7`
If `B = 3/7` , then `A + 3/7 = 1`
`rArr A = 1 - 3/7 = 4/7`
`(x^(2))/((x^(2) - 4) (x^(2) + 3)) = (4)/(7(x^(2) - 4)) + (3)/(7(x^(2) + 3))`
`I = 4/7int(1)/(x^(2)-(2)^(2)) dx + 3/7 int (1)/(x^(2)+(sqrt(3))^(2)) dx`
`= 4/(7).(1)/(2.2) "log" |( x-2)/(x+2)| + 3/(7).(1)/(sqrt(3)) tan^(-1) '(x)/(sqrt(3)) + C`
`= 1/7"log"|(x-2)/(x+2)| + (sqrt(3))/(7) tan^(-1) '(x)/(sqrt(3)) + C`