The integral of \( \cos(3x) e^{\sin(3x)} \) can be evaluated using substitution.

Let:

\[
u = \sin(3x)
\]
Then, the derivative of \( u \) with respect to \( x \) is:
\[
\frac{du}{dx} = 3\cos(3x) \quad \Rightarrow \quad du = 3\cos(3x) dx \quad \Rightarrow \quad \frac{du}{3} = \cos(3x) dx
\]

Now, the integral becomes:

\[
\int \cos(3x) e^{\sin(3x)} dx = \frac{1}{3} \int e^u du
\]

The integral of \( e^u \) is simply \( e^u \), so:

\[
\frac{1}{3} \int e^u du = \frac{1}{3} e^u + C
\]

Finally, substituting back \( u = \sin(3x) \), we get:

\[
\frac{1}{3} e^{\sin(3x)} + C
\]

So the solution is:

\[
\int \cos(3x) e^{\sin(3x)} dx = \frac{1}{3} e^{\sin(3x)} + C
\]

Would you like further details on this solution or have any questions?

Here are 5 related questions:
1. How would you evaluate the integral of \( \cos(kx) e^{\sin(kx)} \) for any constant \( k \)?
2. What is the derivative of \( e^{\sin(3x)} \)?
3. Can you compute the integral \( \int e^{x^2} dx \) using similar methods?
4. How would you handle the integral \( \int \sin(3x) e^{\sin(3x)} dx \)?
5. What is the general rule for integrating exponential functions involving trigonometric terms?

**Tip**: Substitution is a powerful technique for solving integrals, especially when the integrand includes a function and its derivative!

Learn how to evaluate the integral of cos(3x) e^sin(3x) using substitution. This detailed solution covers the substitution method, where u = sin(3x), leading to the simplified integral of an exponential function. Suitable for university-level calculus students.

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