Question #179758

Why integral t2(-e-t) with limits t= 0, ∞ becomes zero.


1
Expert's answer
2021-05-07T09:58:10-0400

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Solution


0t2(et)dt\boxed{\int_0^\infin t^2(-e^{-t})dt}


0t2(et)dt-\int_0^\infin t^2(e^{-t})dt


integrate by parts: 

fg=fgfg∫ f g ′ = f g − ∫ f ′ g


f=t2,g=etstepsf=2t,g=etf = t^ 2 , g ′ = e ^{− t}\\ ↓ steps \\ f ′ = 2 t , g = − e ^{− t}



=t2et2tetdt= − t^ 2 e ^{− t} − ∫ − 2 t e ^{− t} d t


Now solving:

2tetdt{\displaystyle\int}-2t\mathrm{e}^{-t}\,\mathrm{d}t

Apply linearity:

2tetdt-2{\displaystyle\int}t\mathrm{e}^{-t}\,\mathrm{d}t


Now solving:

tetdt{\displaystyle\int}t\mathrm{e}^{-t}\,\mathrm{d}t

Integrate by parts:


fg=fgfg{\int}\mathtt{f}\mathtt{g}' = \mathtt{f}\mathtt{g} - {\int}\mathtt{f}'\mathtt{g}


f=t,g=etstepsf=1,g=et:f = t , g ′ = e ^{− t}\\ ↓ steps \\ f ′ = 1 , g = − e ^{− t} :


=tetetdt=-t\mathrm{e}^{-t}-{\displaystyle\int}-\mathrm{e}^{-t}\,\mathrm{d}t

Now solving:

etdt{\displaystyle\int}-\mathrm{e}^{-t}\,\mathrm{d}t

Substitute u=tdudt=1(steps)dt=du:Substitute \space u = − t ⟶ \frac{ d u} {d t} = − 1 (steps) ⟶ d t = − d u :

=eudu={\displaystyle\int}\mathrm{e}^u\,\mathrm{d}u


on solving


=et=\mathrm{e}^{-t}


Plug in solved integrals:


tetetdt-t\mathrm{e}^{-t}-{\displaystyle\int}-\mathrm{e}^{-t}\,\mathrm{d}t

=tetet=-t\mathrm{e}^{-t}-\mathrm{e}^{-t}


Plug in solved integrals:

2tetdt-2{\displaystyle\int}t\mathrm{e}^{-t}\,\mathrm{d}t

=2tet+2et=2t\mathrm{e}^{-t}+2\mathrm{e}^{-t}

Plug in solved integrals:


t2et2tetdt-t^2\mathrm{e}^{-t}-{\displaystyle\int}-2t\mathrm{e}^{-t}\,\mathrm{d}t

=t2et2tet2et=-t^2\mathrm{e}^{-t}-2t\mathrm{e}^{-t}-2\mathrm{e}^{-t}

Plug in solved integrals:

t2etdt-{\displaystyle\int}t^2\mathrm{e}^{-t}\,\mathrm{d}t

=t2et+2tet+2et=t^2\mathrm{e}^{-t}+2t\mathrm{e}^{-t}+2\mathrm{e}^{-t}


The problem is solved:

t2etdt{\displaystyle\int}-t^2\mathrm{e}^{-t}\,\mathrm{d}t


=(t2+2t+2)et+C=\left(t^2+2t+2\right)\mathrm{e}^{-t}+C


After solving integral , we got



    \implies t2+2t+2et\boxed{t^2+2t+2\over e^t} limit 0 to \infin


When we put limits


And solve



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Limit t_{\rightarrow \infin} [t2+2t+2ett^2+2t+2\over e^t ]


Limit is \infin\over\infin



To solve this


We use L hospital rule , and



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We will get the value , answer is

"0"





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