int_(1/2) ^1 (dx/(xsqrt(4x^2-1))) এর মান কত? 

Explanation:

Another Explanation (5): ```html

প্রশ্ন: \( \int_{1/2}^{1} \frac{dx}{x\sqrt{4x^2-1}} \) এর মান নির্ণয় করো।

সমাধান:

ধরি, \( x = \frac{1}{2} \sec{\theta} \). 🤔 তাহলে, \( dx = \frac{1}{2} \sec{\theta} \tan{\theta} \, d\theta \). 🤓 যখন \( x = \frac{1}{2} \), তখন \( \frac{1}{2} = \frac{1}{2} \sec{\theta} \Rightarrow \sec{\theta} = 1 \Rightarrow \theta = 0 \). 🥳 যখন \( x = 1 \), তখন \( 1 = \frac{1}{2} \sec{\theta} \Rightarrow \sec{\theta} = 2 \Rightarrow \theta = \frac{\pi}{3} \). 🤩 সুতরাং, \( \int_{1/2}^{1} \frac{dx}{x\sqrt{4x^2-1}} = \int_{0}^{\pi/3} \frac{\frac{1}{2} \sec{\theta} \tan{\theta} \, d\theta}{\frac{1}{2} \sec{\theta} \sqrt{4(\frac{1}{4}\sec^2{\theta})-1}} \) \( = \int_{0}^{\pi/3} \frac{\frac{1}{2} \sec{\theta} \tan{\theta} \, d\theta}{\frac{1}{2} \sec{\theta} \sqrt{\sec^2{\theta}-1}} \) \( = \int_{0}^{\pi/3} \frac{\frac{1}{2} \sec{\theta} \tan{\theta} \, d\theta}{\frac{1}{2} \sec{\theta} \tan{\theta}} \) \( = \int_{0}^{\pi/3} d\theta \) \( = [\theta]_{0}^{\pi/3} \) \( = \frac{\pi}{3} - 0 \) \( = \frac{\pi}{3} \). 🎉 অতএব, \( \int_{1/2}^{1} \frac{dx}{x\sqrt{4x^2-1}} = \frac{\pi}{3} \). 😎 ```