What is the integral of+e^{-5x}sin(x) ?

The integral of+e^{-5x}sin(x) is -(e^{-5x}cos(x))/(26)-(5e^{-5x}sin(x))/(26)+C

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integral of+e^{-5x}sin(x)

\int + e^{- 5 x} sin (x) d x

- \frac{e ^{- 5 x} cos ( x )}{26} - \frac{5 e ^{- 5 x} sin ( x )}{26} + C

Solution steps

\int e^{- 5 x} sin (x) d x

Apply Integration By Parts

= - e^{- 5 x} cos (x) - \int 5 e^{- 5 x} cos (x) d x

Take the constant out:

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

= - e^{- 5 x} cos (x) - 5 \cdot \int e^{- 5 x} cos (x) d x

Apply Integration By Parts

= - e^{- 5 x} cos (x) - 5 (e^{- 5 x} sin (x) - \int - 5 e^{- 5 x} sin (x) d x)

Take the constant out:

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

= - e^{- 5 x} cos (x) - 5 (e^{- 5 x} sin (x) - (- 5 \cdot \int e^{- 5 x} sin (x) d x))

Therefore

\int e^{- 5 x} sin (x) d x = - e^{- 5 x} cos (x) - 5 (e^{- 5 x} sin (x) - (- 5 \cdot \int e^{- 5 x} sin (x) d x))

Isolate

\int e^{- 5 x} sin (x) d x

= - \frac{e ^{- 5 x} cos ( x )}{26} - \frac{5 e ^{- 5 x} sin ( x )}{26}

Add a constant to the solution

= - \frac{e ^{- 5 x} cos ( x )}{26} - \frac{5 e ^{- 5 x} sin ( x )}{26} + C

What is the integral of+e^{-5x}sin(x) ?
The integral of+e^{-5x}sin(x) is -(e^{-5x}cos(x))/(26)-(5e^{-5x}sin(x))/(26)+C