What is the integral from 0 to pi/2 of e^{-st}sin(t) ?

The integral from 0 to pi/2 of e^{-st}sin(t) is (-e^{-(pis)/2}s+1)/(s^2+1)

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integral from 0 to pi/2 of e^{-st}sin(t)

\int_{0}^{\frac{π}{2}} e^{- s t} sin (t) d t

\frac{- e ^{- \frac{π s}{2}} s + 1}{s ^{2} + 1}

Solution steps

\int_{0}^{\frac{π}{2}} e^{- s t} sin (t) d t

Compute the indefinite integral:

\int e^{- s t} sin (t) d t = - \frac{e ^{- s t} cos ( t )}{1 + s ^{2}} - \frac{s e ^{- s t} sin ( t )}{1 + s ^{2}} + C

Compute the boundaries:

\int_{0}^{\frac{π}{2}} e^{- s t} sin (t) d t = - \frac{e ^{- \frac{π s}{2}} s}{s ^{2} + 1} - (- \frac{1}{1 + s ^{2}})

= - \frac{e ^{- \frac{π s}{2}} s}{s ^{2} + 1} - (- \frac{1}{1 + s ^{2}})

Simplify

= \frac{- e ^{- \frac{π s}{2}} s + 1}{s ^{2} + 1}

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What is the integral from 0 to pi/2 of e^{-st}sin(t) ?
The integral from 0 to pi/2 of e^{-st}sin(t) is (-e^{-(pis)/2}s+1)/(s^2+1)