Question

Calculo 1. alguem responde por favorrrr

Answer 1

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

******************************************************************

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

*****************************************************************

∫ 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

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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