What is the integral of sin(wt)e^{-st} ?

The integral of sin(wt)e^{-st} is -(we^{-st}cos(wt))/(w^2+s^2)-(se^{-st}sin(wt))/(w^2+s^2)+C

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integral of sin(wt)e^{-st}

\int sin (wt) e^{- s t} d t

- \frac{w e ^{- s t} cos ( wt )}{w ^{2} + s ^{2}} - \frac{s e ^{- s t} sin ( wt )}{w ^{2} + s ^{2}} + C

Solution steps

\int sin (wt) e^{- s t} d t

Apply Integration By Parts

: - e^{- s t} \frac{1}{w} cos (wt) - \int s e^{- s t} \frac{1}{w} cos (wt) d t

= - e^{- s t} \frac{1}{w} cos (wt) - \int s e^{- s t} \frac{1}{w} cos (wt) d t

Take the constant out:

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

= - e^{- s t} \frac{1}{w} cos (wt) - s \frac{1}{w} \cdot \int e^{- s t} cos (wt) d t

Simplify

s \frac{1}{w} : \frac{s}{w}

= - e^{- s t} \frac{1}{w} cos (wt) - \frac{s}{w} \cdot \int e^{- s t} cos (wt) d t

Apply Integration By Parts

: e^{- s t} \frac{1}{w} sin (wt) - \int - s e^{- s t} \frac{1}{w} sin (wt) d t

= - e^{- s t} \frac{1}{w} cos (wt) - \frac{s}{w} (e^{- s t} \frac{1}{w} sin (wt) - \int - s e^{- s t} \frac{1}{w} sin (wt) d t)

Take the constant out:

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

= - e^{- s t} \frac{1}{w} cos (wt) - \frac{s}{w} (e^{- s t} \frac{1}{w} sin (wt) - (- s \frac{1}{w} \cdot \int e^{- s t} sin (wt) d t))

Simplify

- s \frac{1}{w} : - \frac{s}{w}

= - e^{- s t} \frac{1}{w} cos (wt) - \frac{s}{w} (e^{- s t} \frac{1}{w} sin (wt) - (- \frac{s}{w} \cdot \int e^{- s t} sin (wt) d t))

Therefore

\int e^{- s t} sin (wt) d t = - e^{- s t} \frac{1}{w} cos (wt) - \frac{s}{w} (e^{- s t} \frac{1}{w} sin (wt) - (- \frac{s}{w} \cdot \int e^{- s t} sin (wt) d t))

Isolate

\int e^{- s t} sin (wt) d t

= - \frac{w e ^{- s t} cos ( wt )}{w ^{2} + s ^{2}} - \frac{s e ^{- s t} sin ( wt )}{w ^{2} + s ^{2}}

Add a constant to the solution

= - \frac{w e ^{- s t} cos ( wt )}{w ^{2} + s ^{2}} - \frac{s e ^{- s t} sin ( wt )}{w ^{2} + s ^{2}} + C

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What is the integral of sin(wt)e^{-st} ?
The integral of sin(wt)e^{-st} is -(we^{-st}cos(wt))/(w^2+s^2)-(se^{-st}sin(wt))/(w^2+s^2)+C