What is the integral of 3e^{-3x}sin(3x) ?

The integral of 3e^{-3x}sin(3x) is 3(-(e^{-3x}cos(3x))/6-(e^{-3x}sin(3x))/6)+C

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integral of 3e^{-3x}sin(3x)

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

3 (- \frac{e ^{- 3 x} cos ( 3 x )}{6} - \frac{e ^{- 3 x} sin ( 3 x )}{6}) + C

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Therefore

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

Isolate

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

= 3 (- \frac{e ^{- 3 x} cos ( 3 x )}{6} - \frac{e ^{- 3 x} sin ( 3 x )}{6})

Add a constant to the solution

= 3 (- \frac{e ^{- 3 x} cos ( 3 x )}{6} - \frac{e ^{- 3 x} sin ( 3 x )}{6}) + C

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What is the integral of 3e^{-3x}sin(3x) ?
The integral of 3e^{-3x}sin(3x) is 3(-(e^{-3x}cos(3x))/6-(e^{-3x}sin(3x))/6)+C