যদি  int_1^0 1/(1+3x²)=A হয়, তবে A=?

Explanation:

Another Explanation (5): সমাধান: ধরি, \(I = \int_0^1 \frac{1}{1+3x^2} dx\) এখন, \(3x^2 = ( \sqrt{3}x )^2\) সুতরাং, \(I = \int_0^1 \frac{1}{1+(\sqrt{3}x)^2} dx\) আমরা জানি, \(\int \frac{1}{1+x^2} dx = \tan^{-1}(x) + C\) সুতরাং, \(I = \frac{1}{\sqrt{3}} \tan^{-1}(\sqrt{3}x) \Big|_0^1\) 🤔 \(I = \frac{1}{\sqrt{3}} [ \tan^{-1}(\sqrt{3} \cdot 1) - \tan^{-1}(\sqrt{3} \cdot 0) ]\) 🤓 \(I = \frac{1}{\sqrt{3}} [ \tan^{-1}(\sqrt{3}) - \tan^{-1}(0) ]\) 🤩 আমরা জানি, \(\tan^{-1}(\sqrt{3}) = \frac{\pi}{3}\) এবং \(\tan^{-1}(0) = 0\) সুতরাং, \(I = \frac{1}{\sqrt{3}} \cdot \frac{\pi}{3}\) 🥰 \(I = \frac{\pi}{3\sqrt{3}}\) 🎉 অতএব, \(A = \frac{\pi}{3\sqrt{3}}\)