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Properties of Inequalities

∫ 3+4 x-x 2 x...

Question

$\int \frac{3 + 4 x - x^{2}}{(x + 2) (x - 1)} d x$

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Solution

$\int (\frac{3 + 4 x - x^{2}}{x^{2} - x + 2 x - 2}) d x = \int (\frac{- x^{2} + 4 x + 3}{x^{2} + x - 2}) d x \frac{- x^{2} + 4 x + 3}{x^{2} + x - 2} = - 1 + \frac{5 x + 1}{x^{2} + x - 2} . . . (1) ∴ \frac{5 x + 1}{x^{2} + x - 2} = \frac{5 x + 1}{x^{2} + 2 x - x - 2} = \frac{5 x + 1}{x (x + 2) - 1 (x + 2)} Let \frac{5 x + 1}{(x - 1) (x + 2)} = \frac{A}{x - 1} + \frac{B}{x + 2} \Rightarrow \frac{5 x + 1}{(x - 1) (x + 2)} = \frac{A (x + 2) + B (x - 1)}{(x - 1) (x + 2)} \Rightarrow 5 x + 1 = A (x + 2) + B (x - 1) . . . (2) Putting x + 2 = 0 or x = - 2 in eq (2) \Rightarrow 5 x - 2 + 1 = A \times 0 + B (- 2 - 1) \Rightarrow B = 3 Putting x - 1 = 0 or x = 1 in eq (2) \Rightarrow 5 \times 1 + 1 = A (3) + B \times 0 \Rightarrow A = 2 ∴ \frac{5 x + 1}{(x - 1) (x + 2)} = \frac{2}{x - 1} + \frac{3}{x + 2} . . . (3) From (1) & (3) \frac{- x^{2} + 4 x + 3}{x^{2} + x - 2} = - 1 + \frac{2}{x - 1} + \frac{3}{x + 2} \Rightarrow \int \frac{- x^{2} + 4 x + 3}{x^{2} + x - 2} d x = \int - 1 d x + \int \frac{2}{x - 1} d x + \int \frac{3}{x + 2} d x = - x + 2 \ln (x - 1) + 3 \ln (x + 2) + C$

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