int_1^(e^2)  dx/(x(1+lnx)^2  =?

দেওয়া আছে, \( \int_{1}^{e^2} \frac{dx}{x(1+\ln x)^2} \)

ধরি, \( 1 + \ln x = z \)

তাহলে, \( \frac{1}{x} dx = dz \)

যখন \( x = 1 \), তখন \( z = 1 + \ln 1 = 1 + 0 = 1 \)

যখন \( x = e^2 \), তখন \( z = 1 + \ln e^2 = 1 + 2 = 3 \)

সুতরাং, \( \int_{1}^{e^2} \frac{dx}{x(1+\ln x)^2} = \int_{1}^{3} \frac{dz}{z^2} \)

\( = \int_{1}^{3} z^{-2} dz \)

\( = \left[ \frac{z^{-1}}{-1} \right]_{1}^{3} \)

\( = \left[ -\frac{1}{z} \right]_{1}^{3} \)

\( = -\frac{1}{3} - (-\frac{1}{1}) \)

\( = -\frac{1}{3} + 1 \)

\( = \frac{-1+3}{3} \)

\( = \frac{2}{3} \) 🎉