Question

Integral e^x cosx/2 dx alguém me ajuda,please

Answer 1

Resposta:

Se for  ∫ e^(x) * cos(x/2) dx

Fazendo por partes

u= cos(x/2)    ==>du = (1/2) * [-sen(x/2)] dx

dv=e^(x) dx ==> ∫   dv = ∫  e^(x) dx ==> v= e^(x)

∫ e^(x) * cos(x/2) dx =e^(x) *cos(x/2)  - ∫ e^(x)  (1/2) * [-sen(x/2)] dx

∫ e^(x) * cos(x/2) dx =e^(x) *cos(x/2) + (1/2) ∫ e^(x)*sen(x/2) dx   (i)

resolvendo por partes  ∫ e^(x)*sen(x/2) dx

u=sen(x/2)  ==>du =(1/2) * cos(x/2) dx

dv=e^(x) dx ==> ∫   dv = ∫  e^(x) dx ==> v= e^(x)

∫ e^(x)*sen(x/2) dx =e^(x) * sen(x/2)  -  ∫ e^(x) (1/2) * cos(x/2) dx

∫ e^(x)*sen(x/2) dx =e^(x) * sen(x/2)  -  (1/2)∫ e^(x)* cos(x/2) dx   (ii)

(ii)  em (i)

∫ e^(x) * cos(x/2) dx =e^(x) *cos(x/2) + (1/2) [e^(x) * sen(x/2)  -  (1/2)∫ e^(x)* cos(x/2) dx]

(5/4)∫ e^(x) * cos(x/2) dx =e^(x) *cos(x/2) + (1/2) *e^(x) * sen(x/2)  + c

∫ e^(x) * cos(x/2) dx =(4/5)*e^(x) *cos(x/2) + (2/5) [e^(x) * sen(x/2) ] + c