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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.

Evaluate the integral ∫ 6e^x cos(5x) dx

Learn how to evaluate the integral ∫ 6e^x cos(5x) dx with a step-by-step solution. This problem covers advanced integration techniques, including exponential and trigonometric product integration. Suitable for university-level calculus students.

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