What is the integral of e^{-x}sin(nx) ?

The integral of e^{-x}sin(nx) is -(ne^{-x}cos(nx))/(n^2+1)-(e^{-x}sin(nx))/(n^2+1)+C

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integral of e^{-x}sin(nx)

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

- \frac{n e ^{- x} cos ( n x )}{n ^{2} + 1} - \frac{e ^{- x} sin ( n x )}{n ^{2} + 1} + C

Solution steps

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Therefore

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

Isolate

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

= - \frac{n e ^{- x} cos ( n x )}{n ^{2} + 1} - \frac{e ^{- x} sin ( n x )}{n ^{2} + 1}

Add a constant to the solution

= - \frac{n e ^{- x} cos ( n x )}{n ^{2} + 1} - \frac{e ^{- x} sin ( n x )}{n ^{2} + 1} + C

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What is the integral of e^{-x}sin(nx) ?
The integral of e^{-x}sin(nx) is -(ne^{-x}cos(nx))/(n^2+1)-(e^{-x}sin(nx))/(n^2+1)+C