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

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

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

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

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

Solution steps

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

Simplify

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

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

Apply Integration By Parts

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

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

Apply Integration By Parts

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

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

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