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

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

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integral of-e^{-x}sin(e^{-x})

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

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

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

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

Apply u-substitution

: \int sin (v) d v

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

Use the common integral:

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

= - (- (- cos (v)))

Substitute back

= - (- (- cos (e^{- x})))

Simplify

= - cos (e^{- x})

Add a constant to the solution

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

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