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Integration of Irrational Algebraic Fractions - 1

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Question

Evaluate the following integrals:

$\int (x - 3) \sqrt{x^{2} + 3 x - 18} d x$

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Solution

$Let I = \int (x - 3) \sqrt{x^{2} + 3 x - 18} d x We express x - 3 = A (\frac{d}{d x} (x^{2} + 3 x - 18)) + B x - 3 = A (2 x + 3) + B Equating the coefficients of x and constants, we get 1 = 2 A and - 3 = 3 A + B or A = \frac{1}{2} and B = - \frac{9}{2} ∴ I = \int (\frac{1}{2} (2 x + 3) - \frac{9}{2}) \sqrt{x^{2} + 3 x - 18} d x = \frac{1}{2} \int (2 x + 3) \sqrt{x^{2} + 3 x - 18} d x - \frac{9}{2} \int \sqrt{x^{2} + 3 x - 18} d x = \frac{1}{2} I_{1} - \frac{9}{2} I_{2} . . . (1) Now, I_{1} = \int (2 x + 3) \sqrt{x^{2} + 3 x - 18} d x Let x^{2} + 3 x - 18 = u On differentiating both sides, we get (2 x + 3) d x = d u ∴ I_{1} = \int \sqrt{u} d u = \frac{2}{3} u^{\frac{3}{2}} + c_{1} = \frac{2}{3} {(x^{2} + 3 x - 18)}^{\frac{3}{2}} + c_{1} . . . (2) And, I_{2} = \int \sqrt{x^{2} + 3 x - 18} d x = \int \sqrt{x^{2} + 3 x + \frac{9}{4} - \frac{9}{4} - 18} d x = \int \sqrt{{(x + \frac{3}{2})}^{2} - {(\frac{9}{2})}^{2}} d x Let (x + \frac{3}{2}) = u On differentiating both sides, we get d x = d u ∴ I_{2} = \int \sqrt{{(u)}^{2} - {(\frac{9}{2})}^{2}} d u = \frac{u}{2} \sqrt{{(u)}^{2} - {(\frac{9}{2})}^{2}} - \frac{1}{2} {(\frac{9}{2})}^{2} \log |u + \sqrt{{(u)}^{2} - {(\frac{9}{2})}^{2}}| + c_{2} = \frac{x + \frac{3}{2}}{2} \sqrt{{(x + \frac{3}{2})}^{2} - {(\frac{9}{2})}^{2}} - \frac{1}{2} {(\frac{9}{2})}^{2} \log |(x + \frac{3}{2}) + \sqrt{{(x + \frac{3}{2})}^{2} - {(\frac{9}{2})}^{2}}| + c_{2} = \frac{2 x + 3}{4} \sqrt{x^{2} + 3 x - 18} - \frac{81}{8} \log |(x + \frac{3}{2}) + \sqrt{x^{2} + 3 x - 18}| + c_{2} . . . (3) From (1), (2) and (3), we get ∴ I = \frac{1}{2} (\frac{2}{3} {(x^{2} + 3 x - 18)}^{\frac{3}{2}} + c_{1}) - \frac{9}{2} (\frac{2 x + 3}{4} \sqrt{x^{2} + 3 x - 18} - \frac{81}{8} \log |(x + \frac{3}{2}) + \sqrt{x^{2} + 3 x - 18}| + c_{2}) = \frac{1}{3} {(x^{2} + 3 x - 18)}^{\frac{3}{2}} - \frac{9}{8} (2 x + 3) \sqrt{x^{2} + 3 x - 18} + \frac{729}{16} \log |(x + \frac{3}{2}) + \sqrt{x^{2} + 3 x - 18}| + c Hence, \int (x - 3) \sqrt{x^{2} + 3 x - 18} d x = \frac{1}{3} {(x^{2} + 3 x - 18)}^{\frac{3}{2}} - \frac{9}{8} (2 x + 3) \sqrt{x^{2} + 3 x - 18} + \frac{729}{16} \log |(x + \frac{3}{2}) + \sqrt{x^{2} + 3 x - 18}| + c$

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