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प्रश्न
Integrate the rational function:
`1/(x^4 - 1)`
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उत्तर
Let `1/(x^4 - 1) = 1/((x + 1)(x - 1)(x^2 + 1))`
`= A/(x + 1) + B/(x - 1) + (Cx + D)/(x^2 + 1)`
1 ≡ A(x – 1) (x2 + 1) + B(x + 1) (x2 + 1) + (Cx + D) (x + 1) (x – 1) …(1)
Putting x = -1 in equation (1),
1 = A (-1 – 1) (1 + 1)
⇒ 1 = A (-4)
⇒ A = `-1/4`
Putting x = 1 in equation (1),
1 = B (1 + 1) (1 + 1)
⇒ 1= B (2) (2)
⇒ B = `1/4`
Comparing the coefficients of x3 in equation (1),
0 = A + B + C
`=> 0 = (-1)/4 + 1/4 + C`
⇒ C = 0
1 = -A + B - D
`=> 1 = 1/4 + 1/4 - D`
⇒ ` D = -1/2`
`therefore 1/(x^4 - 1) = - 1/(4(x + 1)) + 1/(4(x - 1)) - 1/(2 (x^2 + 1))`
`therefore int dx/(x^4 - 1) = 1/4 int 1/(x + 1) dx + 1/4 int 1/(x - 1) dx - 1/2 int 1/(x^2 + 1) dx`
`= - 1/4 log (x + 1) = 1/4 log (x - 1) -1/2 tan^-1 x + C`
`= 1/4 log ((x - 1)/(x + 1)) - 1/2 tan^-1 x + C`
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