int_0^1 x/(2-x^2)dx  এর মান কত?

সমাধান:

ধরি, \(I = \int_0^1 \frac{x}{2-x^2} dx\)

এখানে, \(2-x^2 = z\) ধরলে, \(-2x dx = dz\) হয়। সুতরাং, \(x dx = -\frac{1}{2} dz\)

যখন \(x = 0\), তখন \(z = 2 - 0^2 = 2\)

যখন \(x = 1\), তখন \(z = 2 - 1^2 = 1\)

অতএব, \(I = \int_2^1 \frac{-\frac{1}{2}}{z} dz = -\frac{1}{2} \int_2^1 \frac{1}{z} dz\)

আমরা জানি, \(\int \frac{1}{z} dz = \ln |z| + C\)

সুতরাং, \(I = -\frac{1}{2} [\ln |z|]_2^1 = -\frac{1}{2} (\ln 1 - \ln 2) = -\frac{1}{2} (0 - \ln 2) = \frac{1}{2} \ln 2\)

অতএব, \(\int_0^1 \frac{x}{2-x^2} dx = \frac{1}{2} \ln 2\) 🥳🎉