The given function is

$f''(x) = e^{2x} \sin(4x).$

To find the integral of $f''(x)$, we calculate $f'(x)$, which is the first antiderivative of $f''(x)$.

Step-by-step integration:

The integral to solve is:

$\int e^{2x} \sin(4x) \, dx.$

This is a standard integral that can be solved using integration by parts or recognizing it as part of the family of exponential and trigonometric products.

Using the formula for exponential and trigonometric integrals:

The general formula for this type of integral is:

$\int e^{ax} \sin(bx) \, dx = \frac{e^{ax}}{a^2 + b^2} \big(a \sin(bx) - b \cos(bx)\big) + C.$

Here:

• $a = 2$,
• $b = 4$.

Substitute these values into the formula:

$\int e^{2x} \sin(4x) \, dx = \frac{e^{2x}}{2^2 + 4^2} \big(2 \sin(4x) - 4 \cos(4x)\big) + C.$

Simplify:

$\int e^{2x} \sin(4x) \, dx = \frac{e^{2x}}{20} \big(2 \sin(4x) - 4 \cos(4x)\big) + C.$

$\int e^{2x} \sin(4x) \, dx = \frac{e^{2x}}{10} \big(\sin(4x) - 2 \cos(4x)\big) + C.$

So, the first integral is:

$f'(x) = \frac{e^{2x}}{10} \big(\sin(4x) - 2 \cos(4x)\big) + C_1.$

Next, to find $f(x)$, you would integrate $f'(x)$:

$f(x) = \int f'(x) \, dx.$

Let me know if you'd like me to proceed with the second integral or if you have any specific questions about this process!

Here are some related questions for deeper understanding:

1. How is the formula for exponential and trigonometric integrals derived?
2. What is the role of integration by parts in solving these types of problems?
3. Can you generalize this integral for $e^{ax} \sin(bx)$ where $a \neq 2, b \neq 4$?
4. How does $C_1$ affect the solution for $f(x)$?
5. What applications do such integrals have in physics or engineering?

Tip: For such problems, memorizing the standard forms of integrals involving $e^{ax} \sin(bx)$ and $e^{ax} \cos(bx)$ can save a lot of time.