int_0^(pi/2)e^x(sinx+cosx)dx=? 

প্রশ্ন: \( \int_{0}^{\pi/2} e^x (\sin x + \cos x) \, dx = ? \)

সমাধান:

আমরা জানি, \( \int e^x [f(x) + f'(x)] \, dx = e^x f(x) + C \)

এখানে, \( f(x) = \sin x \) হলে, \( f'(x) = \cos x \)

সুতরাং, \( \int e^x (\sin x + \cos x) \, dx = e^x \sin x + C \)

এখন, নির্দিষ্ট সমাকল:

\( \int_{0}^{\pi/2} e^x (\sin x + \cos x) \, dx = \left[ e^x \sin x \right]_{0}^{\pi/2} \)

\( = e^{\pi/2} \sin(\pi/2) - e^0 \sin(0) \)

\( = e^{\pi/2} \cdot 1 - 1 \cdot 0 \)

\( = e^{\pi/2} \)

অতএব, \( \int_{0}^{\pi/2} e^x (\sin x + \cos x) \, dx = e^{\pi/2} \)

উত্তর: \( e^{\pi/2} \) 🎉