int_0^(ln(2)) ((e^x)/(e^x+1))dx    এর মান কোনটি  ?  

ধরি, \(I = \int_0^{\ln(2)} \frac{e^x}{e^x + 1} dx\)

এখন, \(e^x + 1 = t\) ধরলে, \(e^x dx = dt\) হবে।

যখন \(x = 0\), তখন \(t = e^0 + 1 = 1 + 1 = 2\)

আবার, যখন \(x = \ln(2)\), তখন \(t = e^{\ln(2)} + 1 = 2 + 1 = 3\)

সুতরাং, \(I = \int_2^3 \frac{1}{t} dt\)

\(= [\ln(t)]_2^3\)

\(= \ln(3) - \ln(2)\)

\(= \ln(\frac{3}{2})\) 🥳

অতএব, \(\int_0^{\ln(2)} \frac{e^x}{e^x + 1} dx = \ln(\frac{3}{2})\) 🥰