What is the integral of e^{-sx}sin(ax) ?

The integral of e^{-sx}sin(ax) is -(ae^{-sx}cos(ax))/(a^2+s^2)-(se^{-sx}sin(ax))/(a^2+s^2)+C

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integral of e^{-sx}sin(ax)

\int e^{- s x} sin (a x) d x

- \frac{a e ^{- s x} cos ( a x )}{a ^{2} + s ^{2}} - \frac{s e ^{- s x} sin ( a x )}{a ^{2} + s ^{2}} + C

Solution steps

\int e^{- s x} sin (a x) d x

Apply Integration By Parts

= - e^{- s x} \frac{1}{a} cos (a x) - \int s e^{- s x} \frac{1}{a} cos (a x) d x

Take the constant out:

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

= - e^{- s x} \frac{1}{a} cos (a x) - s \frac{1}{a} \cdot \int e^{- s x} cos (a x) d x

Simplify

s \frac{1}{a} : \frac{s}{a}

= - e^{- s x} \frac{1}{a} cos (a x) - \frac{s}{a} \cdot \int e^{- s x} cos (a x) d x

Apply Integration By Parts

= - e^{- s x} \frac{1}{a} cos (a x) - \frac{s}{a} (e^{- s x} \frac{1}{a} sin (a x) - \int - s e^{- s x} \frac{1}{a} sin (a x) d x)

Take the constant out:

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

= - e^{- s x} \frac{1}{a} cos (a x) - \frac{s}{a} (e^{- s x} \frac{1}{a} sin (a x) - (- s \frac{1}{a} \cdot \int e^{- s x} sin (a x) d x))

Simplify

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

= - e^{- s x} \frac{1}{a} cos (a x) - \frac{s}{a} (e^{- s x} \frac{1}{a} sin (a x) - (- \frac{s}{a} \cdot \int e^{- s x} sin (a x) d x))

Therefore

\int e^{- s x} sin (a x) d x = - e^{- s x} \frac{1}{a} cos (a x) - \frac{s}{a} (e^{- s x} \frac{1}{a} sin (a x) - (- \frac{s}{a} \cdot \int e^{- s x} sin (a x) d x))

Isolate

\int e^{- s x} sin (a x) d x

= - \frac{a e ^{- s x} cos ( a x )}{a ^{2} + s ^{2}} - \frac{s e ^{- s x} sin ( a x )}{a ^{2} + s ^{2}}

Add a constant to the solution

= - \frac{a e ^{- s x} cos ( a x )}{a ^{2} + s ^{2}} - \frac{s e ^{- s x} sin ( a x )}{a ^{2} + s ^{2}} + C

What is the integral of e^{-sx}sin(ax) ?
The integral of e^{-sx}sin(ax) is -(ae^{-sx}cos(ax))/(a^2+s^2)-(se^{-sx}sin(ax))/(a^2+s^2)+C