Integral definida

por renato formigoni Seg 06 Ago 2012, 18:49

o valor da integral definida ∫_0^(π/2) e^x cos x dx é:

por **Iago6** Seg 06 Ago 2012, 21:24

Integração por partes

$\int_{0}^{\frac{\pi}{2}} e^x cos x = e^x senx - \int e^x senx dx \\\\ = e^x senx - \int (e^x senx )dx \\\\ \int e^x cos x =e^x senx + e^x cos x - \int (e^x cos x )dx \\\\ 2 \int e^x cos x= e^x senx + e^x cos x \\\\ \int_{0}^{\frac{\pi}{2}} e^x cos x= \left [\frac{ e^x senx + e^x cos x}{2} \right ]_{0}^{\frac{\pi}{2}} \\\\ =\left [\frac{ e^{\left (\frac{\pi}{2} \right )} sen\left (\frac{\pi}{2} \right ) + e^{\left (\frac{\pi}{2} \right )} cos\left (\frac{\pi}{2} \right ) }{2} \right ]-\left [\frac{ e^{\left (\frac{\pi}{2} \right )} sen(0) + e^{0} cos(0) }{2} \right ] =\boxed{ {\color{Red} \frac{1}{2}({\color{Red} e^{\frac{\pi}{2}}-1})}}$

Integral definida

Integral definida

Re: Integral definida

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