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

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

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

\int_{0}^{π} cos (t) e^{- s t} d t

\frac{e ^{- π s} s + s}{s ^{2} + 1}

Solution steps

\int_{0}^{π} cos (t) e^{- s t} d t

Compute the indefinite integral:

\int cos (t) e^{- s 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} cos (t) e^{- s 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}

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