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The integral you have provided is:

\[
\int e^{3x} \cos(8x) \, dx
\]

To solve this integral, we can apply **integration by parts** or use the method of **undetermined coefficients**, which is a common approach when dealing with integrals of the form:

\[
\int e^{ax} \cos(bx) \, dx
\]

The general solution for such integrals is of the form:

\[
\int e^{ax} \cos(bx) \, dx = \frac{e^{ax}}{a^2 + b^2} \left(a \cos(bx) + b \sin(bx)\right) + C
\]

For your specific integral, \( a = 3 \) and \( b = 8 \), so the solution becomes:

\[
\int e^{3x} \cos(8x) \, dx = \frac{e^{3x}}{3^2 + 8^2} \left(3 \cos(8x) + 8 \sin(8x)\right) + C
\]

Now let's simplify the constants:

\[
3^2 + 8^2 = 9 + 64 = 73
\]

So the final solution is:

\[
\int e^{3x} \cos(8x) \, dx = \frac{e^{3x}}{73} \left(3 \cos(8x) + 8 \sin(8x)\right) + C
\]

Would you like more details or have any questions?

Here are 5 related questions to explore this topic further:
1. How do you apply integration by parts to solve integrals involving exponentials and trigonometric functions?
2. What is the method of undetermined coefficients, and when is it typically used in integration?
3. What happens if we change the constants in \( e^{ax} \cos(bx) \)? How does that affect the result?
4. How would you approach the integration of \( e^{3x} \sin(8x) \)?
5. What is the physical or geometrical interpretation of integrals involving exponentials and trigonometric functions?

**Tip:** When integrating functions with both exponentials and trigonometric terms, try using formulas or tricks like undetermined coefficients to avoid multiple rounds of integration by parts.

Evaluate the integral \( \int e^{3x} \cos(8x) \, dx \).

Solve the integral \( \int e^{3x} \cos(8x) \, dx \) using the method of undetermined coefficients. This problem involves integration techniques for combining exponential and trigonometric functions. Suitable for university-level calculus students.

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