int_1^2 dx/(5-2x) =?

সমাধান: \( \int_{1}^{2} \frac{dx}{5-2x} \)

ধরি, \( u = 5 - 2x \)

তাহলে, \( du = -2 dx \)

সুতরাং, \( dx = -\frac{1}{2} du \)

যখন \( x = 1 \), তখন \( u = 5 - 2(1) = 3 \)

যখন \( x = 2 \), তখন \( u = 5 - 2(2) = 1 \)

এখন, \( \int_{1}^{2} \frac{dx}{5-2x} = \int_{3}^{1} \frac{-\frac{1}{2} du}{u} \)

\( = -\frac{1}{2} \int_{3}^{1} \frac{du}{u} \)

\( = -\frac{1}{2} [\ln|u|]_{3}^{1} \)

\( = -\frac{1}{2} (\ln|1| - \ln|3|) \)

\( = -\frac{1}{2} (0 - \ln 3) \)

\( = \frac{1}{2} \ln 3 \)

\( = \ln \sqrt{3} \) 🤩