Resposta:
Se for f(x) = x²*√eˣ =x² * e^(x/2)
∫ x² * e^(x/2) dx
Fazendo por Partes
u=x² ==> du=2x dx
dv=e^(x/2) dx ==>∫ dv=∫ e^(x/2) dx ==>v =2* e^(x/2)
∫ x² * e^(x/2) dx = 2*x²* e^(x/2) - ∫ 2* e^(x/2) * 2x dx
∫ x² * e^(x/2) dx = 2*x²* e^(x/2) - 4* ∫x* e^(x/2) dx (1)
calculando ∫x* e^(x/2) dx
u=x ==>du=dx
∫x* e^(x/2) dx =2 * x * e^(x/2) - ∫ 2* e^(x/2) dx
∫x* e^(x/2) dx =2 * x * e^(x/2) - 2 *∫ e^(x/2) dx
∫x* e^(x/2) dx =2 * x * e^(x/2) - 4 e^(x/2) (2)
(2) em (1) ficamos com
∫ x² * e^(x/2) dx = 2*x²* e^(x/2) - 4* [2 * x * e^(x/2) - 4 e^(x/2)] + c
∫ x² * e^(x/2) dx = [2x²-8x +16] * e^(x/2)+ c
∫ x² * e^(x/2) dx = [2x²-8x +16] * √eˣ+ c
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Resposta:
Se for f(x) = x²*√eˣ =x² * e^(x/2)
∫ x² * e^(x/2) dx
Fazendo por Partes
u=x² ==> du=2x dx
dv=e^(x/2) dx ==>∫ dv=∫ e^(x/2) dx ==>v =2* e^(x/2)
∫ x² * e^(x/2) dx = 2*x²* e^(x/2) - ∫ 2* e^(x/2) * 2x dx
∫ x² * e^(x/2) dx = 2*x²* e^(x/2) - 4* ∫x* e^(x/2) dx (1)
calculando ∫x* e^(x/2) dx
Fazendo por Partes
u=x ==>du=dx
dv=e^(x/2) dx ==>∫ dv=∫ e^(x/2) dx ==>v =2* e^(x/2)
∫x* e^(x/2) dx =2 * x * e^(x/2) - ∫ 2* e^(x/2) dx
∫x* e^(x/2) dx =2 * x * e^(x/2) - 2 *∫ e^(x/2) dx
∫x* e^(x/2) dx =2 * x * e^(x/2) - 4 e^(x/2) (2)
(2) em (1) ficamos com
∫ x² * e^(x/2) dx = 2*x²* e^(x/2) - 4* [2 * x * e^(x/2) - 4 e^(x/2)] + c
∫ x² * e^(x/2) dx = [2x²-8x +16] * e^(x/2)+ c
∫ x² * e^(x/2) dx = [2x²-8x +16] * √eˣ+ c
Letra C