int^(e²)₁dx/(x(1+lnx)^2=?

Explanation:

Another Explanation (5): সমাধান: ধরি, \(I = \int_{1}^{e^2} \frac{dx}{x(1 + \ln x)^2}\) এখানে, \(1 + \ln x = z\) ধরলে, \(\frac{1}{x} dx = dz\) হবে। সুতরাং, যখন \(x = 1\), তখন \(z = 1 + \ln 1 = 1 + 0 = 1\). আবার, যখন \(x = e^2\), তখন \(z = 1 + \ln e^2 = 1 + 2 = 3\). অতএব, \(I = \int_{1}^{3} \frac{dz}{z^2}\) \(I = \int_{1}^{3} z^{-2} dz = \left[ \frac{z^{-1}}{-1} \right]_{1}^{3} = \left[ -\frac{1}{z} \right]_{1}^{3}\) \(I = -\frac{1}{3} - (-\frac{1}{1}) = -\frac{1}{3} + 1 = \frac{-1 + 3}{3} = \frac{2}{3}\) সুতরাং, \(\int_{1}^{e^2} \frac{dx}{x(1 + \ln x)^2} = \frac{2}{3}\) 🥳🎉