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The integral you've provided is:

\[
\int e^{4x} \cos(5x) \, dx
\]

This is a standard integral that can be solved using integration by parts or the method of undetermined coefficients (for integrals of the form \( e^{ax} \cos(bx) \)).

### Solution using the formula for integrals of the form \( \int e^{ax} \cos(bx) \, dx \):

For an integral of the form:

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

the result is:

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

In this case, \( a = 4 \) and \( b = 5 \), so applying the formula:

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

Simplifying the denominator:

\[
\int e^{4x} \cos(5x) \, dx = \frac{e^{4x} \left( 4 \cos(5x) + 5 \sin(5x) \right)}{16 + 25} + C
\]

\[
= \frac{e^{4x} \left( 4 \cos(5x) + 5 \sin(5x) \right)}{41} + C
\]

So, the final answer is:

\[
\boxed{\frac{e^{4x} \left( 4 \cos(5x) + 5 \sin(5x) \right)}{41} + C}
\]

Would you like more details on any part of the solution? Here are some related questions you might find useful:

1. What is the method of undetermined coefficients in integration?
2. How does integration by parts work for such integrals?
3. Can this integral be solved using substitution? Why or why not?
4. What other methods can be used to solve complex integrals?
5. How does Euler's formula help in solving integrals involving sine and cosine?

**Tip**: For integrals involving both exponential and trigonometric functions, the method of undetermined coefficients is often the quickest approach.

Learn how to solve the integral ∫ e^(4x) * cos(5x) dx using the method of undetermined coefficients. This step-by-step solution involves key calculus concepts such as exponential and trigonometric functions.

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