Posted on November 4, 2022November 4, 2022 by 202211041204 Solution to 2002-CE-AMATH-II-4 Find . [Hint: Let .] Roughwork.
.
.]![\begin{aligned} &\quad\, \int_{0}^{\frac{1}{2}}\frac{1}{\sqrt{1-x^2}}\,\mathrm{d}x \\ \dots\enspace & \textrm{let }x=\sin\theta\textrm{ s.t. }\mathrm{d}x=\cos\theta\,\mathrm{d}\theta \enspace\dots \\ & = \int_{x=0}^{x=\frac{1}{2}}\frac{\cos\theta}{\sqrt{1-(\sin\theta )^2}}\,\mathrm{d}\theta \\ & = \int_{x=0}^{x=\frac{1}{2}}\frac{\cos\theta}{\cos\theta}\,\mathrm{d}\theta \\ & = \int_{x=0}^{x=\frac{1}{2}}\mathrm{d}\theta \\ & = \big[\theta \big]\big|_{x=0}^{x=\frac{1}{2}} \\ & = \big[\sin^{-1}x\big]\big|_{0}^{\frac{1}{2}} \\ & = \frac{\pi}{6}\\ \end{aligned}](https://physicspupil.com/wp-content/ql-cache/quicklatex.com-11bfae113bc70418631a5fdbb5c2429e_l3.svg "Rendered by QuickLaTeX.com")
