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The integral provided is:

\[
\int x \cos(5x) \, dx
\]

This is an integral that can be solved using the method of integration by parts. Let's recall the formula for integration by parts:

\[
\int u \, dv = uv - \int v \, du
\]

We need to choose \( u \) and \( dv \) wisely. In this case, we can set:

- \( u = x \) (which implies \( du = dx \))
- \( dv = \cos(5x) \, dx \) (which implies \( v = \frac{1}{5} \sin(5x) \), using the integral of cosine)

Now, applying the integration by parts formula:

\[
\int x \cos(5x) \, dx = u v - \int v \, du
\]

Substituting the values:

\[
\int x \cos(5x) \, dx = x \cdot \frac{1}{5} \sin(5x) - \int \frac{1}{5} \sin(5x) \, dx
\]

Now, we solve the remaining integral:

\[
\int \sin(5x) \, dx = -\frac{1}{5} \cos(5x)
\]

Thus, the expression becomes:

\[
\int x \cos(5x) \, dx = \frac{x}{5} \sin(5x) + \frac{1}{25} \cos(5x) + C
\]

where \( C \) is the constant of integration.

### Final Answer:

\[
\int x \cos(5x) \, dx = \frac{x}{5} \sin(5x) + \frac{1}{25} \cos(5x) + C
\]

Let me know if you would like further clarification, or if you have any questions.

Here are 5 related questions for practice:
1. How would you integrate \( \int x \sin(3x) \, dx \)?
2. Can you solve \( \int x^2 \cos(4x) \, dx \) using the same method?
3. How does integration by parts differ from substitution?
4. Can you use integration by parts to solve \( \int e^x \sin(x) \, dx \)?
5. What is the integral of \( \int \cos(7x) \, dx \)?

**Tip:** When performing integration by parts, always choose \( u \) as the function that simplifies upon differentiation, and \( dv \) as the function whose integral is straightforward.

Evalúe la siguiente integral: ∫ x cos(5x) dx

Learn how to solve the integral ∫ x cos(5x) dx using integration by parts. This problem involves the use of trigonometric integrals and is suitable for students in advanced high school calculus or college-level calculus.

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