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Factorization

∫ 4 x 2-5 d x

Question

$\int \sqrt{4 x^{2} - 5} d x$

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Solution

$\int \sqrt{4 x^{2} - 5} d x = \int \sqrt{4 (x^{2} - \frac{5}{4})} d x = 2 \int \sqrt{x^{2} - {(\frac{\sqrt{5}}{2})}^{2}} d x = 2 [\frac{x}{2} \sqrt{x^{2} - \frac{5}{4}} - \frac{5}{8} \ln |x + \sqrt{x^{2} - \frac{5}{4}}|] + C [∵ \int \sqrt{x^{2} - a^{2}} d x = \frac{1}{2} x \sqrt{x^{2} - a^{2}} - \frac{1}{2} a^{2} \ln |x + \sqrt{x^{2} - a^{2}}| + C] = x \sqrt{x^{2} - \frac{5}{4}} - \frac{5}{4} \ln |x + \sqrt{x^{2} - \frac{5}{4}}| + C$

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