int_1^(e^2)dx/(x(1+lnx)^2=? 

Explanation:

Another Explanation (5): সমাধান: ধরি, \( u = 1 + \ln{x} \)। সুতরাং, \( du = \frac{1}{x} dx \)। 🤩 এখন, \( x = 1 \) হলে, \( u = 1 + \ln{1} = 1 + 0 = 1 \)। এবং \( x = e^2 \) হলে, \( u = 1 + \ln{e^2} = 1 + 2 = 3 \)। সুতরাং, সমাকলনটি হবে: \[ \int_{1}^{e^2} \frac{dx}{x(1+\ln{x})^2} = \int_{1}^{3} \frac{du}{u^2} \] এখন, \( \int \frac{1}{u^2} du = \int u^{-2} du = \frac{u^{-1}}{-1} + C = -\frac{1}{u} + C \)। 🤓 সুতরাং, \[ \int_{1}^{3} \frac{du}{u^2} = \left[-\frac{1}{u}\right]_{1}^{3} = -\frac{1}{3} - (-1) = -\frac{1}{3} + 1 = \frac{2}{3} \] অতএব, \(\int_{1}^{e^2} \frac{dx}{x(1+\ln{x})^2} = \frac{2}{3}\)।🥳