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} 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

\frac{d}{d t} (1 - t^{3})

\frac{\partial}{\partial y} (x arctan (yz))

\int e^{- (x^{2} + 1)} x d x

\frac{d}{d x} (x^{3} - 6 x^{2} + 9 x)

\frac{d}{d x} (3 cos (2 x) + ln (3 x))

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