int_0^1(e^sqrtx)/sqrt(x)dx=?

ধরি, \(t = \sqrt{x}\)

সুতরাং, \(x = t^2\)

অতএব, \(dx = 2t dt\)

যখন \(x = 0\), \(t = \sqrt{0} = 0\)

যখন \(x = 1\), \(t = \sqrt{1} = 1\)

তাহলে, সমাকলনটি হবে:

\(\int_0^1 \frac{e^{\sqrt{x}}}{\sqrt{x}} dx = \int_0^1 \frac{e^t}{t} 2t dt = 2 \int_0^1 e^t dt\)

\(= 2 [e^t]_0^1 = 2 (e^1 - e^0) = 2 (e - 1)\)

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