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}cos(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} cos (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} cos (x)

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

Add a constant to the solution

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