$ I = \int\limits_{0}^{\pi\ /2}(1+sinx)(1+tan^2(\frac{x}{2}))e^xdx $
Ta có $ 1 + sinx = 1 + \dfrac{2tan\dfrac{x}{2}}{1 + tan^2\dfrac{x}{2}}
$\Rightarrow $ f(x) = e^x + e^xtan^2\dfrac{x}{2} + 2e^xtan\dfrac{x}{2} $
$2 I_3 = 2 \int_{0}^{\dfrac{\pi}{2}}e^xtan\dfrac{x}{2}dx $
$ u = tan\dfrac{x}{2} , du = 1/2(1+tan^2\dfrac{x}{2}) $, ${dv = e^xdx ; v = e^x } $
$ 2I_3 = 2e^x tan\dfrac{x}{2} - \int_{0}^{\dfrac{\pi}{2}}(e^x + e^xtan^2\dfrac{x}{2})dx $\Rightarrow $ I = 2e^x tan\dfrac{x}{2}|_0^{\dfrac{\pi}{2}} $