int_1^3 (2x)/(1+x^2) dx  এর মান কত? 

Explanation:

Another Explanation (5): সমাধান: 🤔 ধরি, \(u = 1 + x^2\) তাহলে, \(\frac{du}{dx} = 2x\) 😲 সুতরাং, \(du = 2x \, dx\) 😎 এখন, \(x = 1\) হলে, \(u = 1 + 1^2 = 2\) 🥳 এবং \(x = 3\) হলে, \(u = 1 + 3^2 = 10\) 🤩 অতএব, \(\int_1^3 \frac{2x}{1+x^2} \, dx = \int_2^{10} \frac{1}{u} \, du\) 🤓 আমরা জানি, \(\int \frac{1}{u} \, du = \ln|u| + C\) 🤫 সুতরাং, \(\int_2^{10} \frac{1}{u} \, du = \left[ \ln|u| \right]_2^{10}\) 🫡 \( = \ln|10| - \ln|2|\) 😌 \( = \ln(10) - \ln(2)\) 😌 \( = \ln\left(\frac{10}{2}\right)\) 😇 \( = \ln(5)\) 🎉 সুতরাং, \(\int_1^3 \frac{2x}{1+x^2} \, dx = \ln 5\) 🥰