What is the integral of 27e^{3x}sin(3x) ?

The integral of 27e^{3x}sin(3x) is 27((e^{3x}sin(3x))/6-(e^{3x}cos(3x))/6)+C

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integral of 27e^{3x}sin(3x)

\int 27 e^{3 x} sin (3 x) d x

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

Solution steps

\int 27 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

= 27 \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

= 27 (- \frac{1}{3} e^{3 x} cos (3 x) - \int - e^{3 x} cos (3 x) d x)

Take the constant out:

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

= 27 (- \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

= 27 (- \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

27 \cdot \int e^{3 x} sin (3 x) d x = 27 (- \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

= 27 (\frac{e ^{3 x} sin ( 3 x )}{6} - \frac{e ^{3 x} cos ( 3 x )}{6})

Add a constant to the solution

= 27 (\frac{e ^{3 x} sin ( 3 x )}{6} - \frac{e ^{3 x} cos ( 3 x )}{6}) + C

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What is the integral of 27e^{3x}sin(3x) ?
The integral of 27e^{3x}sin(3x) is 27((e^{3x}sin(3x))/6-(e^{3x}cos(3x))/6)+C