Find ∫ x^4 {(x−1)(x^2+1)} x.

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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 = $\frac{x^2}{2} + x + \frac{1}{2} log|x-1| - \frac{1}{2} tan^{-1} + c$