int (e^x (1+x)/cos^2 (xe^x))  সমান-

Explanation:

Another Explanation (5): সমাধান: 🤔 ধরি, \(xe^x = z\) অতএব, \(\frac{dz}{dx} = \frac{d}{dx}(xe^x)\) \(\Rightarrow \frac{dz}{dx} = x\frac{d}{dx}(e^x) + e^x\frac{d}{dx}(x)\) [Product Rule] \(\Rightarrow \frac{dz}{dx} = xe^x + e^x \cdot 1\) \(\Rightarrow \frac{dz}{dx} = e^x(x+1)\) \(\Rightarrow dz = e^x(1+x)dx\) এখন, প্রদত্ত সমাকল: \(\int \frac{e^x(1+x)}{\cos^2(xe^x)} dx\) = \(\int \frac{1}{\cos^2(z)} dz\) [যেহেতু \(e^x(1+x)dx = dz\) এবং \(xe^x = z\)] = \(\int \sec^2(z) dz\) [যেহেতু \(\frac{1}{\cos(z)} = \sec(z)\)] = \(\tan(z) + c\) [আমরা জানি, \(\int \sec^2(x) dx = \tan(x) + c\)] = \(\tan(xe^x) + c\) [z এর মান বসিয়ে] সুতরাং, \(\int \frac{e^x(1+x)}{\cos^2(xe^x)} dx = \tan(xe^x) + c\) 😍