What is the integral of e^{-7x}sin(x) ?

The integral of e^{-7x}sin(x) is -(e^{-7x}cos(x))/(50)-(7e^{-7x}sin(x))/(50)+C

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integral of e^{-7x}sin(x)

\int e^{- 7 x} sin (x) d x

- \frac{e ^{- 7 x} cos ( x )}{50} - \frac{7 e ^{- 7 x} sin ( x )}{50} + C

Solution steps

\int e^{- 7 x} sin (x) d x

Apply Integration By Parts

= - e^{- 7 x} cos (x) - \int 7 e^{- 7 x} cos (x) d x

Take the constant out:

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

= - e^{- 7 x} cos (x) - 7 \cdot \int e^{- 7 x} cos (x) d x

Apply Integration By Parts

= - e^{- 7 x} cos (x) - 7 (e^{- 7 x} sin (x) - \int - 7 e^{- 7 x} sin (x) d x)

Take the constant out:

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

= - e^{- 7 x} cos (x) - 7 (e^{- 7 x} sin (x) - (- 7 \cdot \int e^{- 7 x} sin (x) d x))

Therefore

\int e^{- 7 x} sin (x) d x = - e^{- 7 x} cos (x) - 7 (e^{- 7 x} sin (x) - (- 7 \cdot \int e^{- 7 x} sin (x) d x))

Isolate

\int e^{- 7 x} sin (x) d x

= - \frac{e ^{- 7 x} cos ( x )}{50} - \frac{7 e ^{- 7 x} sin ( x )}{50}

Add a constant to the solution

= - \frac{e ^{- 7 x} cos ( x )}{50} - \frac{7 e ^{- 7 x} sin ( x )}{50} + C

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What is the integral of e^{-7x}sin(x) ?
The integral of e^{-7x}sin(x) is -(e^{-7x}cos(x))/(50)-(7e^{-7x}sin(x))/(50)+C