int_0^a 1/(sqrt(a^2-x^2)) dx=? 

প্রশ্ন: \(\int_0^a \frac{1}{\sqrt{a^2-x^2}} dx = ?\) 🤔

সমাধান:

আমরা জানি, \(\int \frac{1}{\sqrt{a^2-x^2}} dx = \sin^{-1}(\frac{x}{a}) + C\) 🤓

অতএব, \(\int_0^a \frac{1}{\sqrt{a^2-x^2}} dx = \left[ \sin^{-1}(\frac{x}{a}) \right]_0^a \) 🧐

\(= \sin^{-1}(\frac{a}{a}) - \sin^{-1}(\frac{0}{a}) \)

\(= \sin^{-1}(1) - \sin^{-1}(0) \)

\(= \frac{\pi}{2} - 0 \)

\(= \frac{\pi}{2}\) 🎉

সুতরাং, \(\int_0^a \frac{1}{\sqrt{a^2-x^2}} dx = \frac{\pi}{2}\) ✅