To solve the integral 

\[
\int 6x \arctan(x) \, dx,
\]

we can use integration by parts, which follows the formula:

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

where we need to identify parts of the function that can be used as \( u \) and \( dv \).

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

Let:
- \( u = \arctan(x) \) (since its derivative is simpler)
- \( dv = 6x \, dx \)

### Step 2: Differentiate \( u \) and integrate \( dv \)

- Differentiate \( u \):
  
  \[
  du = \frac{1}{1 + x^2} \, dx
  \]

- Integrate \( dv \):

  \[
  v = \int 6x \, dx = 3x^2
  \]

### Step 3: Apply the integration by parts formula

Substituting into the formula:

\[
\int 6x \arctan(x) \, dx = 3x^2 \arctan(x) - \int 3x^2 \cdot \frac{1}{1 + x^2} \, dx
\]

Simplify the second integral:

\[
\int 3x^2 \cdot \frac{1}{1 + x^2} \, dx = \int \frac{3x^2}{1 + x^2} \, dx
\]

### Step 4: Simplify the second integral

Rewrite the fraction:

\[
\frac{3x^2}{1 + x^2} = 3 - \frac{3}{1 + x^2}
\]

Thus, the integral becomes:

\[
\int \frac{3x^2}{1 + x^2} \, dx = \int 3 \, dx - \int \frac{3}{1 + x^2} \, dx = 3x - 3 \arctan(x)
\]

### Step 5: Substitute back into the expression

Putting it all together:

\[
\int 6x \arctan(x) \, dx = 3x^2 \arctan(x) - (3x - 3 \arctan(x))
\]

Simplify:

\[
\int 6x \arctan(x) \, dx = 3x^2 \arctan(x) - 3x + 3 \arctan(x) + C
\]

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

### Final Answer:

\[
\int 6x \arctan(x) \, dx = 3x^2 \arctan(x) - 3x + 3 \arctan(x) + C
\]

Would you like more details on any step, or do you have any questions?

### Related Questions:
1. How would you solve \(\int x^2 \arctan(x) \, dx\)?
2. What happens if you change \(\arctan(x)\) to \(\arcsin(x)\) in the integral?
3. Can you explain the concept of integration by parts?
4. What is the derivative of \(\arctan(x)\) and why does it look like that?
5. How do you evaluate \(\int \frac{x}{1+x^2} \, dx\)?

### Tip:
When using integration by parts, always choose \( u \) as a function that simplifies when differentiated!

Learn how to solve the integral ∫ 6x arctan(x) dx using integration by parts. This problem involves the arctangent function and demonstrates the step-by-step process of integration by parts, suitable for college-level students.

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