Ta có:
$\int\limits_0^{\frac{\pi}{2}}\dfrac{e^x\sin xdx}{(\sin x+\cos x)^2}$
$=\dfrac{1}{2}\int\limits_0^{\frac{\pi}{2}}\left(\dfrac{e^x(\sin x+\cos x)}{(\sin x+\cos x)^2}+\dfrac{e^x(\sin x-\cos x)}{(\sin x+\cos x)^2}\right)dx$
$=\dfrac{1}{2}\int\limits_0^{\frac{\pi}{2}}\dfrac{1}{\sin x+\cos x}d(e^x)+\dfrac{1}{2}\int\limits_0^{\frac{\pi}{2}}e^xd\left(\dfrac{1}{\sin x+\cos x}\right)$
$=\dfrac{e^x}{2(\sin x+\cos x)}\left|\begin{array}{l}\dfrac{\pi}{2}\\0\end{array}\right.=\dfrac{\sqrt{e^{\pi}-1}}{2}$