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

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

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

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

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

Solution steps

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

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

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

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

Simplify

- (- \frac{e ^{- x} cos ( x )}{2} - \frac{e ^{- x} sin ( x )}{2}) + \frac{e ^{- x} sin ( x )}{2} - \frac{e ^{- x} cos ( 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}sin(x)+e^{-x}cos(x)) ?
The integral of (-e^{-x}sin(x)+e^{-x}cos(x)) is e^{-x}sin(x)+C