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

The integral of cos(x)*e^{-x} is (e^{-x}sin(x))/2-(e^{-x}cos(x))/2+C

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

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

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

Solution steps

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

Apply Integration By Parts

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

Take the constant out:

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

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

Apply Integration By Parts

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

Therefore

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

Isolate

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

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

Add a constant to the solution

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

What is the integral of cos(x)*e^{-x} ?
The integral of cos(x)*e^{-x} is (e^{-x}sin(x))/2-(e^{-x}cos(x))/2+C