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

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

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

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

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

Solution steps

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

Apply u-substitution

= \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)

\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

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