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

The integral of sin(ax)e^{-bx} is -(ae^{-bx}cos(ax))/(a^2+b^2)-(be^{-bx}sin(ax))/(a^2+b^2)+C

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

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

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

Solution steps

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

Apply Integration By Parts

: - e^{- b x} \frac{1}{a} cos (a x) - \int b e^{- b x} \frac{1}{a} cos (a x) d x

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

Simplify

b \frac{1}{a} : \frac{b}{a}

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

Apply Integration By Parts

: e^{- b x} \frac{1}{a} sin (a x) - \int - b e^{- b x} \frac{1}{a} sin (a x) d x

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

Simplify

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

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

Therefore

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

Isolate

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

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

Add a constant to the solution

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

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What is the integral of sin(ax)e^{-bx} ?
The integral of sin(ax)e^{-bx} is -(ae^{-bx}cos(ax))/(a^2+b^2)-(be^{-bx}sin(ax))/(a^2+b^2)+C