int_0^1x²e^(x^3)dx=? 

Explanation:

Another Explanation (5): সমাধান: ধরি, \(I = \int_{0}^{1} x^2 e^{x^3} dx\) এখন, \(x^3 = z\) ধরি। সুতরাং, \(3x^2 dx = dz\) বা, \(x^2 dx = \frac{1}{3} dz\) সীমা পরিবর্তন করি: যখন \(x = 0\), তখন \(z = 0^3 = 0\) যখন \(x = 1\), তখন \(z = 1^3 = 1\) অতএব, \(I = \int_{0}^{1} e^z \frac{1}{3} dz\) \(I = \frac{1}{3} \int_{0}^{1} e^z dz\) \(I = \frac{1}{3} [e^z]_{0}^{1}\) \(I = \frac{1}{3} [e^1 - e^0]\) \(I = \frac{1}{3} [e - 1]\) সুতরাং, \(\int_{0}^{1} x^2 e^{x^3} dx = \frac{1}{3}(e-1)\) 🎉