Mathematics - Definite Integration Question with Solution | TestHub

$\begin{array}{rl}& \sqrt{1-{\left(y^{\prime }(x)\right)}^2}=y(x)\\ & 1-{\left(\frac{dy}{dx}\right)}^2=y^2\\ & {\left(\frac{dy}{dx}\right)}^2=1-y^2\end{array}$ $\begin{array}{rl}& \frac{dy}{\sqrt{1-y^2}}=dx\text{ OR }\frac{dy}{\sqrt{1-y^2}}=-dx\\ & \Rightarrow {\sin }^{-1}y=x+c,{\sin }^{-1}y=-x+c\\ & x=0,y=0\Rightarrow c=0\\ & {\sin }^{-1}y=x,\text{ as }y\ge 0\\ & \sin x=y\\ & \Rightarrow \frac{dy}{dx}=\cos x\\ & \frac{d^{2}y}{d{x}^2}=-\sin x\\ & \Rightarrow -\sin x+\sin x+1=1\end{array}$