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

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

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

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

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

Solution steps

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

Expand

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

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int e^{- x} cos (x) d x = \frac{e ^{- x} sin ( x )}{2} - \frac{e ^{- x} cos ( x )}{2}

\int e^{- x} sin (x) d x = - \frac{e ^{- x} cos ( x )}{2} - \frac{e ^{- x} sin ( x )}{2}

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

Simplify

\frac{e ^{- x} sin ( x )}{2} - \frac{e ^{- x} cos ( x )}{2} - (- \frac{e ^{- x} cos ( x )}{2} - \frac{e ^{- x} sin ( x )}{2}) : e^{- x} sin (x)

= e^{- x} sin (x)

Add a constant to the solution

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

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