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Question
`int x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3)) "d"x`
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Solution
Let I = `int x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3)) "d"x`
Let `x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3))`
= `"A"/(x^2 + 1) + "b"/(x^2 - 2) + "c"/(x^2 + 3)`
∴ x2 = A(x2 − 2)(x2 + 3) + B(x2 + 1)(x2 + 3) + C(x2 + 1)(x2 − 2) ........(i)
Putting x2 = 2 in (i), we get
2 = B × 3 × 5
∴ B = `2/15`
Putting x2 = −3 in (i), we get
−3 = C × (– 2) × (– 5)
∴ C = `(-3)/10`
Putting x2 = −1 in (i), we get
−1 = A × (–3) × 2
∴ A = `1/6`
∴ `x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3)) = (1/6)/(x^2 + 1) + (2/15)/(x^2 - 2) + ((-3)/10)/(x^2 + 3)`
∴ I = `int[1/(6(x^2 + 1)) + 2/(15(x^2 - 2)) - 3/(10(x^2 + 3))] "d"x`
= `1/6 int 1/(x^2 + 1) "d"x + 2/15 int 1/(x^2 - 2) "d"x - 3/10 int 1/(x^2 + 3) "d"x`
= `1/6 int 1/(x^2 + 1) "d"x + 2/15 int 1/(x^2 - (sqrt(2))^2) "d"x - 3/10 int 1/(x^2 + (sqrt(3))^2) "d"x`
= `1/6 tan^-1x + 2/15 xx 1/(2 xx sqrt(2)) log|(x - sqrt(2))/(x + sqrt(2))| - 3/10 xx 1/sqrt(3) tan^-1 (x/sqrt(3)) + "c"`
∴ I = `1/6 tan^-1x + 1/(15sqrt(2)) log|(x - sqrt(2))/(x + sqrt(2))| - sqrt(3)/10 tan^-1 (x/sqrt(3)) + "c"`
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