Explain how calculate f(2) if f(x)=integral sign(0-1)e^-t(t^(x-1))dt, x>1, f(1)=1-(1/e)?

Expert Answers
sciencesolve eNotes educator| Certified Educator

You need to substitute 2 for x in the given function such that:

`f(2) = int_0^1 e^(-t)*t^(2-1) dt`

`f(2) = int_0^1 e^(-t)*t dt`

You need to use integration by parts to evaluate f(2) such that:

`u = t => du = dt`

`dv = e^(-t) dt => v = -e^(-t)`

`int_0^1 e^(-t)*t dt = -te^(-t)|_0^1 + int_0^1 e^(-t) dt`

`int_0^1 e^(-t)*t dt = -te^(-t)|_0^1 - e^(-t)|_0^1`

`int_0^1 e^(-t)*t dt = -e^(-1) - e^(-1) + e^0`

`int_0^1 e^(-t)*t dt = -2e^(-1) + 1`

`int_0^1 e^(-t)*t dt = 1 + 2/e`

Hence, evaluating f(2) under the given conditions yields `f(2) = 1 + 2/e.`

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