int_0^1xe^(x^(2))dx এর মান কত?

Explanation:

Another Explanation (5): সমাধান: ধরি, \(I = \int_0^1 xe^{x^2} dx\) এখানে, \(x^2 = z\) ধরলে, \(2x dx = dz\) সুতরাং, \(x dx = \frac{1}{2} dz\) যখন \(x = 0\), তখন \(z = 0^2 = 0\) এবং যখন \(x = 1\), তখন \(z = 1^2 = 1\) অতএব, \(I = \int_0^1 e^z \frac{1}{2} dz\) \(I = \frac{1}{2} \int_0^1 e^z dz\) \(I = \frac{1}{2} [e^z]_0^1\) \(I = \frac{1}{2} [e^1 - e^0]\) \(I = \frac{1}{2} [e - 1]\) সুতরাং, \(\int_0^1 xe^{x^2} dx = \frac{1}{2}(e-1)\) 🥳