Question

Find the integral of 1/sqrt(4-x^2)

Answer 1

Factor out the 4 from square root. Then take it outside of the integral and you should now immediately recognise this as the derivative of arcsin(x). We know that arcsin(x/2) will have an extra 1/2 at the end because of chain rule so we add a 2 in front of the indefinite integral to cancel it ou. Now you just expand and cancel and write the final answer. \[\int\limits\limits\limits_{}^{}\frac{ 1 }{ \sqrt{4-x^2} }dx=\int\limits_{}^{}\frac{ 1 }{ \sqrt{4(1-\frac{ x^2 }{ 4 }} )}dx=\int\limits_{}^{}\frac{ 1 }{ \sqrt{4} \sqrt{1- (\frac{ x }{ 2 })^2}}dx=\frac{ 1 }{ 2 } \int\limits_{}^{}\frac{ 1 }{ \sqrt{1-(\frac{ x }{ 2})^2} }dx\]\[=\frac{ 1 }{ \cancel{2} } \left[ \cancel{2}\sin^{-1}\left( \frac{ x }{ 2 } \right) \right]=\sin^{-1}\left( \frac{ x }{ 2 } \right)+C\]

@alimon