Respostas
a)
∫x * eˣ dx
u= x du=dx
eˣ dx = dv ==> ∫eˣ dx = ∫dv ==> eˣ = v
∫x * eˣ dx = x * eˣ - ∫ eˣ dx = x * eˣ - eˣ = eˣ *(x-1) + const
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b)
∫x * sen x dx
por partes:
u=x ==> du = dx
sen x dx = dv ==> ∫ sen x dx = ∫ dv ==> -cos x = v
∫x * sen x dx = -x * cos x+ ∫ cos x dx = -x* cos x + sen x + const
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c)
∫ x² eˣ dx
por partes:
u=x² ==> du =2x dx
eˣ dx = dv ==> ∫ eˣ dx =∫ dv ==> eˣ = v
∫ x² eˣ dx = 2x * eˣ - ∫ eˣ * 2x dx = x² * eˣ -2 * ∫ eˣ * x dx
******* ∫ eˣ * x dx = eˣ *(x-1) letra a *******
∫ x² eˣ dx = x² * eˣ -2* eˣ *(x-1) =eˣ *(x²-2x +2) + cost
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d)
∫ x * ln x dx
por partes:
u= ln x ==> du = (1/x) * dx
x dx = dv ==> ∫ x dx = ∫ dv ==> x²/2 =v
∫ x * ln x dx = (x²/2) * ln x - ∫ x²/2 (1/x) * dx
∫ x * ln x dx = (x²/2) * ln x - ∫ x/2 dx =x²/2 * ln x - x²/4 + const
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e)
∫ ln x dx
por partes
u =ln x ==> du = 1/x dx
dx = dv ==> ∫dx = ∫dv == x=v
∫ ln x dx = x* ln x - ∫ x * 1/x dx = x*ln x - x + const
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f)
∫ x * sec²x dx
u=x ==> du = dx
sec²x dx = dv ==>∫ sec² xdx = ∫ dv ==> tan x = v
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∫ sec² x dx
sec² x = 1/cos²x = 1+ sen²x/cos²x = 1 + sen x * sen x/cos²x
∫ 1 + sen x * sen x/cos²x dx
∫ 1 dx + ∫ sen x * sen x/cos²x dx
∫ 1 dx =x
∫ sen x * sen x/cos²x dx
Integrando por partes:
u= sen x ==> du =cos x dx
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dv =sen x/cos²x dx ==> ∫ dv =∫ sen x/cos²x dx
u= cos x ==> du =-sen x dx
∫ sen x/u² du/(-sen x) = - ∫ 1/u² du = u⁻¹/(-1) = 1/u
sendo u = cos x ==> ∫ sen x/cos²x dx = 1/cos x
v = 1/cos x
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∫ sen x * sen x/cos²x dx =(1/cos x) * sen x - ∫ 1/cos x * cos x dx
=tan x -x
∫ sec² x dx = x + tan x -x = tan x
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∫ x * sec²x dx = x * tan x - ∫ tan x dx
******∫ tan x dx = ∫ sen x/cos x dx u = cos x ==> du =-sen x dx
****** ∫ sen x/u * du/(-sen x) = - ∫ (1/u)du =-ln u ==> - ln (cos x)
∫ x * sec²x dx = x * tan x - ∫ tan x dx =x * tan x + ln (cos x) + const
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g)∫x * (ln x)² dx
por partes:
u=(ln x)² ==> du =2*(log x) / x dx
dv = x dx ==> ∫ dv = ∫x dx ==>v =x²/2
∫x * (ln x)² dx = (ln x)² * x²/2 - ∫x²/2 *2*(log x) / x dx
∫x * (ln x)² dx = (ln x)² * x²/2 - ∫ x (log x) dx
***Letra d ==> ∫ x * ln x dx = x²/2 * ln x - x²/4
∫x * (ln x)² dx =(ln x)² * x²/2 - [x²/2 * ln x - x²/4] + const
______________________________________________h)
∫ (ln x)² dx
por partes:
u=(ln x)² ==> du =2*(ln x) / x dx
dv = dx ==> ∫ dv = ∫ dx ==>v =x
∫x * (ln x)² dx = (ln x)² * x²/2 - ∫x *2*(ln x) / x dx
∫x * (ln x)² dx = (ln x)² * x²/2 - 2 ∫ (ln x) dx
####letra e ∫ ln x dx = x*ln x - x + const
∫ (ln x)² dx =(ln x)² * x - 2*[x*ln x - x] + const_
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i)
∫ x³ * cos x²dx u= x² ==> du =2x dx
∫ x³ * cos u * du/2x
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=(1/2) * ∫ u * cos u * du
Fazendo por partes :
t = u ==> dt = du
cos u * du =dv ==>∫cos u * du = ∫ dv
sen u = v
∫ u * cos u * du = u * sen u - ∫ sen u du
∫ u * cos u * du = u * sen u + cos u
= (1/2) * ∫ u * cos u * du = 1/2 * ( u * sen u + cos u) + const
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Como u = x²
∫ x³ * cos x²dx = 1/2 * ( x² * sen x² + cos x²) + const