The problem asks you to evaluate the integral:

$\int 6e^x \cos(5x) \, dx$

To solve this, we can use integration by parts or recognize this as a typical integral of the product of an exponential function and a trigonometric function. The general strategy here involves using the following identity:

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

For the given problem:

• $a = 1$ (since the exponent of $e^x$ is just $x$, or $1x$),
• $b = 5$ (the frequency of cosine is 5).

Applying the formula:

$\int e^x \cos(5x) \, dx = \frac{e^x}{1^2 + 5^2} \left(1 \cdot \cos(5x) + 5 \cdot \sin(5x)\right)$

$= \frac{e^x}{26} \left(\cos(5x) + 5 \sin(5x)\right)$

Since the integral is for $6e^x \cos(5x)$, multiply this by 6:

$\int 6e^x \cos(5x) \, dx = \frac{6e^x}{26} \left(\cos(5x) + 5 \sin(5x)\right)$

Simplify the fraction:

$= \frac{3e^x}{13} \left(\cos(5x) + 5 \sin(5x)\right) + C$

None of the given answer choices exactly match this. However, looking closely at the choices, the closest option is A or B, depending on the exact representation of constants. In this case, the constant 61 might have been introduced from a different interpretation. I would advise double-checking how the constants are handled in your problem context.

Do you want further clarifications or any specific step explained in more detail?

Here are 5 related questions to expand your understanding:

1. How does integration by parts work, and how can it apply here?
2. What is the method for integrating functions of the form $e^{ax} \cos(bx)$?
3. How would you compute the integral if it involved a sine function instead of cosine?
4. Why do we add the constant $C$ after integrating?
5. How can you verify your integration result by differentiating it?

Tip: Whenever you encounter exponential and trigonometric products, consider using formulas or integration by parts to simplify the process.