7.2 definite integral — ગણિત ધોરણ 12 સાયન્સ — Question

$I = \int_{\pi /4}^{\pi /2} {{e^x}\log \sin x\,dx + \int_{\pi /4}^{\pi /2} {{e^x}\cot x\,dx} } $

$ = \int_{\pi /4}^{\pi /2} {{e^x}\log \sin xdx + [{e^x}\log \sin x]_{\pi /4}^{\pi /2}} $$ - \int_{\pi /4}^{\pi /2} {{e^x}\log \sin x\,dx} $

$ = {e^{\pi /2}}\log \sin \frac{\pi }{2} - {e^{\pi /4}}\log \sin \frac{\pi }{4} $

$= \frac{1}{2}{e^{\pi /4}}\log 2$.