int_0^1 x e^(x^2) dx=?

সমাধান:

ধরি, \(u = x^2\)

সুতরাং, \(\frac{du}{dx} = 2x\)

অতএব, \(dx = \frac{du}{2x}\)

যখন \(x = 0\), তখন \(u = 0^2 = 0\)

যখন \(x = 1\), তখন \(u = 1^2 = 1\)

এখন, সমাকলনটি হবে:

\(\int_0^1 x e^{x^2} dx = \int_0^1 x e^u \frac{du}{2x}\)

\(= \frac{1}{2} \int_0^1 e^u du\)

\(= \frac{1}{2} [e^u]_0^1\)

\(= \frac{1}{2} (e^1 - e^0)\)

\(= \frac{1}{2} (e - 1)\)

সুতরাং, \(\int_0^1 x e^{x^2} dx = \frac{1}{2}(e-1)\) 🎉