What is the integral of sin(3x-2)e^{2x} ?

The integral of sin(3x-2)e^{2x} is -(3e^{2x}cos(3x-2))/(13)+(2e^{2x}sin(3x-2))/(13)+C

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integral of sin(3x-2)e^{2x}

\int sin (3 x - 2) e^{2 x} d x

- \frac{3 e ^{2 x} cos ( 3 x - 2 )}{13} + \frac{2 e ^{2 x} sin ( 3 x - 2 )}{13} + C

Solution steps

\int sin (3 x - 2) e^{2 x} d x

Apply Integration By Parts

: - \frac{1}{3} e^{2 x} cos (3 x - 2) - \int - \frac{2}{3} e^{2 x} cos (3 x - 2) d x

= - \frac{1}{3} e^{2 x} cos (3 x - 2) - \int - \frac{2}{3} e^{2 x} cos (3 x - 2) d x

Take the constant out:

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

= - \frac{1}{3} e^{2 x} cos (3 x - 2) - (- \frac{2}{3} \cdot \int e^{2 x} cos (3 x - 2) d x)

Apply Integration By Parts

: \frac{1}{3} e^{2 x} sin (3 x - 2) - \int \frac{2}{3} e^{2 x} sin (3 x - 2) d x

= - \frac{1}{3} e^{2 x} cos (3 x - 2) - (- \frac{2}{3} (\frac{1}{3} e^{2 x} sin (3 x - 2) - \int \frac{2}{3} e^{2 x} sin (3 x - 2) d x))

Take the constant out:

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

= - \frac{1}{3} e^{2 x} cos (3 x - 2) - (- \frac{2}{3} (\frac{1}{3} e^{2 x} sin (3 x - 2) - \frac{2}{3} \cdot \int e^{2 x} sin (3 x - 2) d x))

Therefore

\int e^{2 x} sin (3 x - 2) d x = - \frac{1}{3} e^{2 x} cos (3 x - 2) - (- \frac{2}{3} (\frac{1}{3} e^{2 x} sin (3 x - 2) - \frac{2}{3} \cdot \int e^{2 x} sin (3 x - 2) d x))

Isolate

\int e^{2 x} sin (3 x - 2) d x

= - \frac{3 e ^{2 x} cos ( 3 x - 2 )}{13} + \frac{2 e ^{2 x} sin ( 3 x - 2 )}{13}

Add a constant to the solution

= - \frac{3 e ^{2 x} cos ( 3 x - 2 )}{13} + \frac{2 e ^{2 x} sin ( 3 x - 2 )}{13} + C

\int \frac{1}{( 2 + x ) ^{3}} d x

y^{^{''}} + 2 y = sin (2 x)

\frac{d}{d x} (\frac{x ^{4} - 3 x ^{2} + 1}{( 2 x + 3 ) ^{4}})

\frac{d}{d x} (\frac{2 x}{2})

4 x^{2} + 16 y^{2} - 51

What is the integral of sin(3x-2)e^{2x} ?
The integral of sin(3x-2)e^{2x} is -(3e^{2x}cos(3x-2))/(13)+(2e^{2x}sin(3x-2))/(13)+C