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

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

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

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

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

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply u-substitution

: \int cos (v) d v

= - \int cos (v) d v

Use the common integral:

\int cos (v) d v = sin (v)

= - sin (v)

Substitute back

= - sin (e^{- x})

Add a constant to the solution

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

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