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Solution
∫0∞t2(−e−t)dt
−∫0∞t2(e−t)dt
integrate by parts:
∫fg′=fg−∫f′g
f=t2,g′=e−t↓stepsf′=2t,g=−e−t
=−t2e−t−∫−2te−tdt
Now solving:
∫−2te−tdt
Apply linearity:
−2∫te−tdt
Now solving:
∫te−tdt
Integrate by parts:
∫fg′=fg−∫f′g
f=t,g′=e−t↓stepsf′=1,g=−e−t:
=−te−t−∫−e−tdt
Now solving:
∫−e−tdt
Substitute u=−t⟶dtdu=−1(steps)⟶dt=−du:
=∫eudu
on solving
=e−t
Plug in solved integrals:
−te−t−∫−e−tdt
=−te−t−e−t
Plug in solved integrals:
−2∫te−tdt
=2te−t+2e−t
Plug in solved integrals:
−t2e−t−∫−2te−tdt
=−t2e−t−2te−t−2e−t
Plug in solved integrals:
−∫t2e−tdt
=t2e−t+2te−t+2e−t
The problem is solved:
∫−t2e−tdt
=(t2+2t+2)e−t+C
After solving integral , we got
⟹ ett2+2t+2 limit 0 to ∞
When we put limits
And solve
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Limit t→∞ [ett2+2t+2 ]
Limit is ∞∞
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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