What is the integral of (1+e^{2x})sin(3x) ?

The integral of (1+e^{2x})sin(3x) is -1/3 cos(3x)-(3e^{2x}cos(3x))/(13)+(2e^{2x}sin(3x))/(13)+C

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integral of (1+e^{2x})sin(3x)

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

- \frac{1}{3} cos (3 x) - \frac{3 e ^{2 x} cos ( 3 x )}{13} + \frac{2 e ^{2 x} sin ( 3 x )}{13} + C

Solution steps

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

Expand

(1 + e^{2 x}) sin (3 x) : sin (3 x) + e^{2 x} sin (3 x)

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

= - \frac{1}{3} cos (3 x) - \frac{3 e ^{2 x} cos ( 3 x )}{13} + \frac{2 e ^{2 x} sin ( 3 x )}{13}

Add a constant to the solution

= - \frac{1}{3} cos (3 x) - \frac{3 e ^{2 x} cos ( 3 x )}{13} + \frac{2 e ^{2 x} sin ( 3 x )}{13} + C

\int x sec (8) x^{2} tan (8) x^{2} d x

\frac{d}{d x} (tan (3 x))

\frac{d}{d x} (\frac{( 8 x )}{x ^{2} + 4})

x \to 0 lim (\frac{1 - e ^{x}}{x})

f (x) = 1 - 3 x^{[2]}, at x = 0

What is the integral of (1+e^{2x})sin(3x) ?
The integral of (1+e^{2x})sin(3x) is -1/3 cos(3x)-(3e^{2x}cos(3x))/(13)+(2e^{2x}sin(3x))/(13)+C