To integrate \( e^{-5x} \cos(2x) \, dx \), we can use the method of **integration by parts** or apply a formula for integrating products of exponentials and trigonometric functions. A direct way is to use the formula for integrals of the form:

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

Here, \( a = -5 \) and \( b = 2 \). Plugging these values into the formula:

\[
\int e^{-5x} \cos(2x) \, dx = \frac{e^{-5x}}{(-5)^2 + 2^2} \left( -5 \cos(2x) + 2 \sin(2x) \right) + C
\]

Simplifying:

\[
\int e^{-5x} \cos(2x) \, dx = \frac{e^{-5x}}{25 + 4} \left( -5 \cos(2x) + 2 \sin(2x) \right) + C
\]

\[
\int e^{-5x} \cos(2x) \, dx = \frac{e^{-5x}}{29} \left( -5 \cos(2x) + 2 \sin(2x) \right) + C
\]

This is the final result.

Would you like more details on integration by parts or this formula? Here are 5 related questions to expand on this:

1. How do you apply integration by parts to this problem?
2. What is the general solution to the integral of an exponential times a sine function?
3. How would this change if the integral was \( e^{-5x} \sin(2x) \, dx \)?
4. What is the physical significance of this integral in engineering applications?
5. Can you solve this problem using a different method?

**Tip:** Knowing standard integrals can save time and simplify complex calculations!

Learn how to integrate e^(-5x) cos(2x) using the standard formula for products of exponentials and trigonometric functions. This detailed solution includes the formula and step-by-step simplification, perfect for calculus students at the undergraduate level.

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