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

The integral of-(cos(x))/(e^x) is -1/2 e^{-x}sin(x)+1/2 e^{-x}cos(x)+C

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

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

- \frac{1}{2} e^{- x} sin (x) + \frac{1}{2} e^{- x} cos (x) + C

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

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

\int \frac{sin ( x )}{e ^{x}} d x = - \frac{cos ( x )}{e ^{x}} - \frac{1}{2} e^{- x} sin (x) + \frac{1}{2} e^{- x} cos (x)

= - (- \frac{cos ( x )}{e ^{x}} - (- \frac{cos ( x )}{e ^{x}} - \frac{1}{2} e^{- x} sin (x) + \frac{1}{2} e^{- x} cos (x)))

Simplify

- (- \frac{cos ( x )}{e ^{x}} - (- \frac{cos ( x )}{e ^{x}} - \frac{1}{2} e^{- x} sin (x) + \frac{1}{2} e^{- x} cos (x))) : - \frac{1}{2} e^{- x} sin (x) + \frac{1}{2} e^{- x} cos (x)

= - \frac{1}{2} e^{- x} sin (x) + \frac{1}{2} e^{- x} cos (x)

Add a constant to the solution

= - \frac{1}{2} e^{- x} sin (x) + \frac{1}{2} e^{- x} cos (x) + C

What is the integral of-(cos(x))/(e^x) ?
The integral of-(cos(x))/(e^x) is -1/2 e^{-x}sin(x)+1/2 e^{-x}cos(x)+C