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Question

Mathematics Question on integral

Integrate the function: 4−x2\sqrt{4-x^2}

Answer

LetI=∫4−x2dx=∫(2)2−(x)2dxI = \int \sqrt{4-x^2}dx = \int \sqrt{(2)^2-(x)^2}dx

It is known that,∫a2−x2dx=x2a2−x2a22sin⁡−1xx+C\int \sqrt {a^2-x^2}dx = \frac{x}{2} \sqrt {a^2-x^2} \frac{a^2}{2}\sin^{-1}\frac{x}{x}+C

∴I=x24−x2+42sin⁡−1 x2+CI = \frac{x}{2}\sqrt{4-x^2}+\frac{4}{2}\sin^{-1}\ \frac{x}{2}+C

=x24−x2+2sin⁡−1x2+C\frac{x}{2}\sqrt{4-x^2}+2\sin^{-1}\frac{x}{2}+C