int_0^(π/4)(sec²xdx)/(1+tanx) এর মান কোনটি?

Explanation:

Another Explanation (5): সমাধান: ধরি, \(I = \int_{0}^{\frac{\pi}{4}} \frac{\sec^2 x}{1 + \tan x} dx\) এখন, \(1 + \tan x = z\) ধরলে, \(\sec^2 x dx = dz\) হয়। সুতরাং, যখন \(x = 0\), তখন \(z = 1 + \tan 0 = 1 + 0 = 1\)। এবং যখন \(x = \frac{\pi}{4}\), তখন \(z = 1 + \tan \frac{\pi}{4} = 1 + 1 = 2\)। তাহলে, সমাকলনটি হবে: \(I = \int_{1}^{2} \frac{1}{z} dz\) আমরা জানি, \(\int \frac{1}{x} dx = \ln |x| + C\)। অতএব, \(I = [\ln |z|]_{1}^{2} = \ln |2| - \ln |1| = \ln 2 - 0 = \ln 2\) সুতরাং, \(\int_{0}^{\frac{\pi}{4}} \frac{\sec^2 x}{1 + \tan x} dx = \ln 2\) 🥳