What is the integral from 1 to 2 of e^{-st}(t) ?

The integral from 1 to 2 of e^{-st}(t) is (-2se^{-2s}-e^{-2s}+se^{-s}+e^{-s})/(s^2)

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integral from 1 to 2 of e^{-st}(t)

\int_{1}^{2} e^{- s t} (t) d t

\frac{- 2 s e ^{- 2 s} - e ^{- 2 s} + s e ^{- s} + e ^{- s}}{s ^{2}}

Solution steps

\int_{1}^{2} e^{- s t} t d t

Apply u-substitution

: \int_{- s}^{- s \cdot 2} \frac{e ^{u} u}{s ^{2}} d u

= \int_{- s}^{- s \cdot 2} \frac{e ^{u} u}{s ^{2}} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{s ^{2}} \cdot \int_{- s}^{- s \cdot 2} e^{u} u d u

Apply Integration By Parts

: [e^{u} u - \int e^{u} d u]_{- s}^{- 2 s}

= \frac{1}{s ^{2}} [e^{u} u - \int e^{u} d u]_{- s}^{- 2 s}

\int e^{u} d u = e^{u}

= \frac{1}{s ^{2}} [e^{u} u - e^{u}]_{- s}^{- 2 s}

Simplify

\frac{1}{s ^{2}} [e^{u} u - e^{u}]_{- s}^{- 2 s} : \frac{[ e ^{u} u - e ^{u} ] _{- s}^{- 2 s}}{s ^{2}}

= \frac{[ e ^{u} u - e ^{u} ] _{- s}^{- 2 s}}{s ^{2}}

Compute the boundaries:

- 2 s e^{- 2 s} - e^{- 2 s} + s e^{- s} + e^{- s}

= \frac{- 2 s e ^{- 2 s} - e ^{- 2 s} + s e ^{- s} + e ^{- s}}{s ^{2}}

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What is the integral from 1 to 2 of e^{-st}(t) ?
The integral from 1 to 2 of e^{-st}(t) is (-2se^{-2s}-e^{-2s}+se^{-s}+e^{-s})/(s^2)