integrate: 3x+5/[x3-x2-x+1]

integrate: 3x+5/[x³-x²-x+1]

Let I = \int \frac{(3 x + 5)}{(x^{3} - x^{2} - x + 1)} dx = \int \frac{(3 x + 5)}{x^{2} (x - 1) - 1 (x - 1)} dx = \int \frac{(3 x + 5) dx}{(x - 1) (x^{2} - 1)} = \int \frac{(3 x + 5) dx}{{(x - 1)}^{2} (x + 1)} Let \frac{3 x + 5}{{(x - 1)}^{2} (x + 1)} = \frac{A}{x - 1} + \frac{B}{{(x - 1)}^{2}} + \frac{C}{x + 1} Now, 3 x + 5 = A (x^{2} - 1) + B (x + 1) + C {(x - 1)}^{2} \Rightarrow 3 x + 5 = {Ax}^{2} - A + Bx + B + {Cx}^{2} + C - 2 Cx \Rightarrow 3 x + 5 = (A + C) x^{2} + (B - 2 C) x + (- A + B + C) Comparing the coefficients of x^{2} on both sides, we get A + C = 0 . . . . . (1) comparing the coefficient of x on both sides, we get B - 2 C = 3 . . . . . (2) Comparing the constants on both sides, we get - A + B + C = 5 . . . . (3) solving (1), (2) and (3), we get A = - \frac{1}{2}; C = \frac{1}{2}; B = 4 so, \frac{3 x + 5}{{(x - 1)}^{2} (x + 1)} = - \frac{1}{2} \times \frac{1}{x - 1} + \frac{4}{{(x - 1)}^{2}} + \frac{1}{2} \times \frac{1}{x + 1} \Rightarrow \int \frac{(3 x + 5) dx}{{(x - 1)}^{2} (x + 1)} = - \frac{1}{2} \int \frac{dx}{x - 1} + 4 \int \frac{dx}{{(x - 1)}^{2}} + \frac{1}{2} \int \frac{dx}{x + 1} = - \frac{1}{2} \log |x - 1| - \frac{4}{x - 1} + \frac{1}{2} \log |x + 1| + C