int_0^1 x/(2-x^2)dx এর মান কত?

সমাধান:

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

এখন, \(2-x^2 = z\) ধরলে, \(-2x dx = dz\)

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

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

\(= -\frac{1}{2} [0 - \ln 2] = \frac{1}{2} \ln 2\)

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

সুতরাং, নির্ণেয় মান \(\frac{1}{2} \log_e 2\) 🥳