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

A. 2(e-1)

B. e+1

C. e/2

D. 1/2(e-1)

সঠিক উত্তরঃ D. 1/2(e-1)

Explanation:

Another Explanation (5): ```html

ধরি, \(u = x^2\)

তাহলে, \(\frac{du}{dx} = 2x\)

সুতরাং, \(du = 2x dx\)

বা, \(x dx = \frac{1}{2} du\)

এখন, যখন \(x = 0\), তখন \(u = 0^2 = 0\)

এবং যখন \(x = 1\), তখন \(u = 1^2 = 1\)

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

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

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

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

\(= \frac{1}{2} (e - 1)\) 🎉

অতএব, \(\int_0^1 xe^{x^2} dx = \frac{1}{2}(e-1)\) ✅

```

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