What is the integral of (sin(3x))/(e^x) ?

The integral of (sin(3x))/(e^x) is -(3e^{-x}cos(3x))/(10)-(e^{-x}sin(3x))/(10)+C

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integral of (sin(3x))/(e^x)

\int \frac{sin ( 3 x )}{e ^{x}} d x

- \frac{3 e ^{- x} cos ( 3 x )}{10} - \frac{e ^{- x} sin ( 3 x )}{10} + C

Solution steps

\int \frac{sin ( 3 x )}{e ^{x}} d x

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

\frac{sin ( 3 x )}{e ^{x}} = (sin (3 x)) e^{- x}

= \int sin (3 x) e^{- x} d x

Apply Integration By Parts

: - \frac{1}{3} e^{- x} cos (3 x) - \int \frac{1}{3} e^{- x} cos (3 x) d x

= - \frac{1}{3} e^{- x} cos (3 x) - \int \frac{1}{3} e^{- x} cos (3 x) d x

Take the constant out:

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

= - \frac{1}{3} e^{- x} cos (3 x) - \frac{1}{3} \cdot \int e^{- x} cos (3 x) d x

Apply Integration By Parts

: \frac{1}{3} e^{- x} sin (3 x) - \int - \frac{1}{3} e^{- x} sin (3 x) d x

= - \frac{1}{3} e^{- x} cos (3 x) - \frac{1}{3} (\frac{1}{3} e^{- x} sin (3 x) - \int - \frac{1}{3} e^{- x} sin (3 x) d x)

Take the constant out:

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

= - \frac{1}{3} e^{- x} cos (3 x) - \frac{1}{3} (\frac{1}{3} e^{- x} sin (3 x) - (- \frac{1}{3} \cdot \int e^{- x} sin (3 x) d x))

Therefore

\int e^{- x} sin (3 x) d x = - \frac{1}{3} e^{- x} cos (3 x) - \frac{1}{3} (\frac{1}{3} e^{- x} sin (3 x) - (- \frac{1}{3} \cdot \int e^{- x} sin (3 x) d x))

Isolate

\int e^{- x} sin (3 x) d x

= - \frac{3 e ^{- x} cos ( 3 x )}{10} - \frac{e ^{- x} sin ( 3 x )}{10}

Add a constant to the solution

= - \frac{3 e ^{- x} cos ( 3 x )}{10} - \frac{e ^{- x} sin ( 3 x )}{10} + C

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What is the integral of (sin(3x))/(e^x) ?
The integral of (sin(3x))/(e^x) is -(3e^{-x}cos(3x))/(10)-(e^{-x}sin(3x))/(10)+C