What is the integral of (sin(4e^{-x}))/(e^x) ?

The integral of (sin(4e^{-x}))/(e^x) is 1/4 cos(4e^{-x})+C

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integral of (sin(4e^{-x}))/(e^x)

\int \frac{sin ( 4 e ^{- x} )}{e ^{x}} d x

\frac{1}{4} cos (4 e^{- x}) + C

Solution steps

\int \frac{sin ( 4 e ^{- x} )}{e ^{x}} d x

Apply u-substitution

: \int - sin (4 u) d u

= \int - sin (4 u) d u

Take the constant out:

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

= - \int sin (4 u) d u

Apply u-substitution

: \int sin (v) \frac{1}{4} d v

= - \int sin (v) \frac{1}{4} d v

Take the constant out:

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

= - \frac{1}{4} \cdot \int sin (v) d v

Use the common integral:

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

= - \frac{1}{4} (- cos (v))

Substitute back

= - \frac{1}{4} (- cos (4 e^{- x}))

Simplify

= \frac{1}{4} cos (4 e^{- x})

Add a constant to the solution

= \frac{1}{4} cos (4 e^{- x}) + C

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What is the integral of (sin(4e^{-x}))/(e^x) ?
The integral of (sin(4e^{-x}))/(e^x) is 1/4 cos(4e^{-x})+C