int _0^ln2 e^x/(1+e^x)dx=?

Explanation:

Another Explanation (5): সমাধান: ধরি, \(I = \int_{0}^{\ln 2} \frac{e^x}{1+e^x} dx\) এখন, \(1+e^x = u\) ধরি। সুতরাং, \(e^x dx = du\) হবে। সীমা পরিবর্তন করি: যখন \(x = 0\), \(u = 1+e^0 = 1+1 = 2\) যখন \(x = \ln 2\), \(u = 1+e^{\ln 2} = 1+2 = 3\) তাহলে, \(I = \int_{2}^{3} \frac{1}{u} du\) 😃 \(I = [\ln |u|]_{2}^{3}\) ✨ \(I = \ln 3 - \ln 2\) 🤩 \(I = \ln \frac{3}{2}\) ✅ অতএব, \(\int_{0}^{\ln 2} \frac{e^x}{1+e^x} dx = \ln \frac{3}{2}\) 💯