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

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

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

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

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

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Therefore

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

Isolate

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

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

Add a constant to the solution

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

\int_{0}^{7 r} 49 r^{2} - y^{2} d y

\int ln (1 + r^{2}) r d r

\frac{\partial}{\partial x} (yz ln (x y))

y (x + y + 1) d x + (x + 2 y) d y = 0

x^{^{''}} - 2 x - 5 = 0, x (0) = 1, x (1) = 3

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