What is the integral of e^{-2x}*sin(e^{-x}) ?

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

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integral of e^{-2x}*sin(e^{-x})

\int e^{- 2 x} \cdot sin (e^{- x}) d x

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

Solution steps

\int e^{- 2 x} sin (e^{- x}) d x

Apply u-substitution

: \int - u sin (u) d u

= \int - u sin (u) d u

Take the constant out:

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

= - \int u sin (u) d u

Apply Integration By Parts

: - u cos (u) - \int - cos (u) d u

= - (- u cos (u) - \int - cos (u) d u)

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

= - (- u cos (u) - (- sin (u)))

Substitute back

u = e^{- x}

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

Simplify

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

Add a constant to the solution

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

x \to 0 lim (x^{\frac{2}{3}})

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