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

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

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integral from 0 to pi of e^{-st}cos(t)

\int_{0}^{π} e^{- s t} cos (t) d t

\frac{e ^{- π s} s + s}{s ^{2} + 1}

Solution steps

\int_{0}^{π} e^{- s t} cos (t) d t

Compute the indefinite integral:

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

Compute the boundaries:

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

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

Simplify

= \frac{e ^{- π s} s + s}{s ^{2} + 1}

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