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

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

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

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

- e^{- x} sin (x) + C

Solution steps

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

Apply u-substitution

= \int cos (ln (u)) + sin (ln (u)) d u

Apply Integration By Parts

= u (cos (ln (u)) + sin (ln (u))) - \int - sin (ln (u)) + cos (ln (u)) d u

\int - sin (ln (u)) + cos (ln (u)) d u = u cos (ln (u))

= u (cos (ln (u)) + sin (ln (u))) - u cos (ln (u))

Substitute back

u = e^{- x}

= e^{- x} (cos (ln (e^{- x})) + sin (ln (e^{- x}))) - e^{- x} cos (ln (e^{- x}))

Simplify

e^{- x} (cos (ln (e^{- x})) + sin (ln (e^{- x}))) - e^{- x} cos (ln (e^{- x})) : - e^{- x} sin (x)

= - e^{- x} sin (x)

Add a constant to the solution

= - e^{- x} sin (x) + C

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