Integrals — Class 12 Maths Solution

Let $I = \int {\sqrt {4 - {x^2}} dx} = \int {\sqrt {{{\left( 2 \right)}^2} - {x^2}} dx}$

$= \left[ {\cfrac{x}{2}\sqrt {{{\left( 2 \right)}^2} - {x^2}} + \cfrac{4}{2}{{\sin }^{ - 1}}\left( {\cfrac{x}{2}} \right)} \right] + C$

$= \cfrac{{x\sqrt {4 - {x^2}} }}{2} + \cfrac{4}{2}{\sin ^{ - 1}}\left( {\cfrac{x}{2}} \right) + C = \cfrac{{x\sqrt {4 - {x^2}} }}{2} + 2{\sin ^{ - 1}}\left( {\cfrac{x}{2}} \right) + C$