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Question
Write a value of\[\int\sqrt{4 - x^2} \text{ dx }\]
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Solution
\[\int \sqrt{4 - x^2} dx\]
\[ = \int \sqrt{2^2 - x^2} \text{ dx }\]
\[ = \frac{x}{2}\sqrt{2^2 - x^2} + \frac{2^2}{2} \sin^{- 1} \left( \frac{x}{2} \right) + C \left( \because \sqrt{a^2 - x^2} = \frac{x}{2}\sqrt{a^2 - x^2} - \frac{a^2}{2} \sin^{- 1} \frac{x}{a} + C \right)\]
\[ = \frac{x}{2}\sqrt{4 - x^2} + 2 \sin^{- 1} \left( \frac{x}{2} \right) + C\]
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