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.