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Find \(\int\) \(\frac{X^4}{{(x−1)(x^2+1)}}\) x.

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⇒ A(x2 + 1) + (Bx + C)( − 1) = 1 

⇒ (A + B)x2 + (C − B)x + A − C = 1. 

⇒ A + B = 0, C − B = 0 and A − C = 1. (By comparing the coefficients of x2 , x and constant ) 

⇒ A + B + C − B = 0 + 0 

⇒ A + C = 0 and A − C = 1. 

⇒ A + C + A − C = 0 + 1 

⇒ 2A = 1 

⇒ A = \(\frac{1}{2}\)

Now,A + B= 0 and A − C = 1 

⇒ B = −A = − \(\frac{1}{2}\) and C = A − 1 = \(\frac{1}{2}\) − 1 = − \(\frac{1}{2}\) .

Hence, \(\int \frac{x^4}{(x+ 1)(x^2+1)}\)dx = \(\frac{x^2}{2} + x + \frac{1}{2} log|x-1| - \frac{1}{2} tan^{-1} + c\)

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