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The integral you provided is:

\[
\int x(x^2 + 1)^5 \, dx
\]

To solve this using integration by parts, we need to identify the parts of the integrand that correspond to \( u \) and \( dv \). 

### Step 1: Choose \( u \) and \( dv \)

Let's choose:
- \( u = x \)
- \( dv = (x^2 + 1)^5 \, dx \)

### Step 2: Compute \( du \) and \( v \)

Now, we differentiate \( u \) and integrate \( dv \):

1. \( du = dx \)
2. To find \( v \), we integrate \( dv \):
   \[
   v = \int (x^2 + 1)^5 \, dx
   \]

The integral of \( (x^2 + 1)^5 \) would require another method, such as substitution. However, since the expression for \( v \) is complicated and the integration by parts approach isn't straightforward here, a better approach is using substitution directly.

### Substitution Method

Instead, let's use the substitution method.

Let \( u = x^2 + 1 \), so:
- \( du = 2x \, dx \)
- \( \frac{1}{2} du = x \, dx \)

The integral becomes:
\[
\int x(x^2 + 1)^5 \, dx = \frac{1}{2} \int u^5 \, du
\]

### Step 3: Integrate

Now, integrate:
\[
\frac{1}{2} \int u^5 \, du = \frac{1}{2} \cdot \frac{u^6}{6} + C = \frac{1}{12} u^6 + C
\]

### Step 4: Substitute back

Finally, substitute back \( u = x^2 + 1 \) to get:
\[
\frac{1}{12} (x^2 + 1)^6 + C
\]

### Summary
- **\( u \) in substitution**: \( x^2 + 1 \)
- **\( v \) after substitution**: \(\frac{(x^2 + 1)^6}{12}\)
- **\( du \)**: \( 2x \, dx \)
- **\( dv \) or \( dx \) used**: \( \frac{1}{2} \)

Would you like further details or have any other questions?

Here are five related questions to explore further:

1. How do you solve an integral using substitution in general?
2. What are the advantages of substitution over integration by parts in certain integrals?
3. Can you use integration by parts when the integral contains a power of a polynomial?
4. How would the solution change if the integrand were \( x(x^2 + 2)^5 \) instead?
5. What methods can you use to check the correctness of an integral's solution?

**Tip:** When you see a polynomial raised to a power, substitution is often the quickest way to simplify the integral.

Learn how to solve the integral x(x^2 + 1)^5 using a combination of substitution and integration by parts. This detailed solution covers key integration methods, including substitution and integration by parts, and demonstrates their application to complex integrals. Suitable for advanced undergraduate students and anyone interested in calculus methods.

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