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Question
Write a value of\[\int\sqrt{x^2 - 9} \text{ dx}\]
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Solution
\[\int \sqrt{x^2 - 9} \text{ dx }\]
\[ = \int \sqrt{x^2 - 3^2} \text{ dx}\]
\[ = \frac{x}{2}\sqrt{x^2 - 3^2} - \frac{3^2}{2}\text{ ln} \left| x + \sqrt{x^2 - 3} \right| + C \left( \because \sqrt{x^2 - a^2} = \frac{x}{2}\sqrt{x^2 - a^2} - \frac{a^2}{2}\text{ ln } \left| x + \sqrt{x^2 + a^2} \right| + C \right)\]
\[ = \frac{x}{2}\sqrt{x^2 - 9} - \frac{9}{2}\text{ ln } \left| x + \sqrt{x^2 - 9} \right| + C\]
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