[Toán 12] Giúp mình giải bài tích phân

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dien0709

$ 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}} $
 
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