int_0^(ln2) e^x/(1+e^x) dx = ? 

Explanation:

Another Explanation (5): সমাধান: ধরি, \(u = 1 + e^x\) তাহলে, \(\frac{du}{dx} = e^x\) অথবা, \(du = e^x dx\) এখন, যখন \(x = 0\), তখন \(u = 1 + e^0 = 1 + 1 = 2\) এবং যখন \(x = \ln 2\), তখন \(u = 1 + e^{\ln 2} = 1 + 2 = 3\) সুতরাং, \(\int_0^{\ln 2} \frac{e^x}{1 + e^x} dx = \int_2^3 \frac{1}{u} du\) \(= [\ln |u|]_2^3\) \(= \ln |3| - \ln |2|\) \(= \ln 3 - \ln 2\) \(= \ln \frac{3}{2}\) 🥳 অতএব, \(\int_0^{\ln 2} \frac{e^x}{1 + e^x} dx = \ln \frac{3}{2}\) 🤩