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

The integral of cos(e^{-x}) is -\Ci(e^{-x})+C

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integral of cos(e^{-x})

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

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

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{cos ( v )}{v} d v

= - \int \frac{cos ( v )}{v} d v

Use the nonelementary integral:

\int \frac{cos ( x )}{x} d x = Ci (x)

= - Ci (v)

Substitute back

= - Ci (e^{- x})

Add a constant to the solution

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

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