Question

integral t^2*e^5t dt

Answer 1

por partes
int udv = u.v - int v.du
u=t²    dv = e^(5t)dt
du = 2tdt          v = (1/5)(e^5t)

int (t².e^(5t))dt = t² .(1/5)(e^5t)   - int (1/5)e^(5t)2tdt 

int (1/5)(e^5t)2tdt : de novo por partes:
u=2t           dv = (1/5)(e^(5t))
du=2.dt      v = (1/25)(e^(5t))
int (1/5)(e^5t)2tdt  = (2t/25)(e^(5t))  -  int (2/25)(e^(5t))dt
substuindo essa integral na de cima:
int (t².e^(5t))dt = t² .(1/5)(e^5t)   - int (1/5)e^(5t)2tdt 
int (t².e^(5t))dt = t² .(1/5)(e^5t)   -[ (2t/25)(e^(5t))  -  int (2/25)(e^(5t))dt]
t² .(1/5)(e^5t)   - (2t/25)e^(5t) + int (2/25)(e^(5t))dt    
                                               essa é uma integral simples de resolver:(joga constante pra fora:
t² .(1/5)(e^5t)   - (2t/25)e^(5t) + (2/25) int (e^(5t))dt   
int e^(5t) = (1/5)e^(5t) então a resposta fica:
t² .(1/5)(e^5t)   - (2t/25)e^(5t)+ (2/25) (1/5)e^(5t)
t² .(1/5)(e^5t)  - (2t/25)e^(5t) + (2/125)e^(5t)       + C

Answer 2

Olá!

Temos:

∫t².e^5tdt --> Faremos por Integração por partes.

Fazendo u = t², vem:

du/dt = 2t => du = 2tdt

E ainda:

dv = e^5tdt => ∫dv = ∫e^5tdt => v = ∫e^5tdt 

---------------------------------------------------------------------------------------------------

v = ∫e^5tdt 

Fazendo n = 5t, vem:

dn/dt = 5 => dn = 5dt => dt = dn/5

v = ∫eⁿ.dn/5 = 1/5.∫eⁿdn = 1/5.eⁿ = 1/5.e^5t

----------------------------------------------------------------------------------------------------

Voltando:

v = ∫e^5tdt = 1/5.e^5t

Logo:

∫udv = uv - ∫vdu 

Substituindo:

∫t².e^5tdt = t².1/5.e^5t - ∫1/5.e^5t.2tdt = 1/5.t².e^5t - 1/5.∫e^5t.2tdt 

∫t².e^5tdt = 1/5.t².e^5t - 1/5.A

----------------------------------------------------------------------------------------------------

A = ∫e^5t.2tdt --> Por partes:

u = 2t => du/dt = 2 => du = 2dt

dv = e^5tdt => v = 1/5.e^5t

Mais uma vez:

∫udv = uv - ∫vdu 

∫2t.e^5tdt = 1/5.2t.e^5t - 1/5.∫e^5t.2dt

∫e^5t.2tdt = 1/5.2t.e^5t - 2/5.∫e^5tdt = 1/5.2t.e^5t - 2/5.1/5e^5t 

∫e^5t.2tdt = 1/5.2t.e^5t - 2/25.e^5t  = 2(1/5t.e^5t - 1/25e^5t) =  A

----------------------------------------------------------------------------------------------------

Voltando, tínhamos:

∫t².e^5tdt = 1/5.t².e^5t - 1/5.A --> Substituindo A:

∫t².e^5tdt = 1/5.t².e^5t - 1/5.[2(1/5t.e^5t - 1/25.e^5t)]

∫t²e^5tdt = 1/5.t².e^5t - 2/5.(1/5.t.e^5t - 1/25.e^5t) + K

Espero realmente ter ajudado! :)