∫ x² arctan(x) dx

Steps to Solve

1. Identify the parts for integration by parts

For the integral $\int x^2 \arctan(x) , dx$, we will use integration by parts, where we identify $u$ and $dv$.

Let:

• $u = \arctan(x)$, so $du = \frac{1}{1+x^2} , dx$
• $dv = x^2 , dx$, so $v = \frac{x^3}{3}$

2. Apply the integration by parts formula

The integration by parts formula is given by:

$$ \int u , dv = uv - \int v , du $$

Substituting our identified parts:

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

3. Simplify the second integral

The second integral becomes:

$$ \int \frac{x^3}{3(1+x^2)} , dx $$

This can be rewritten as:

$$ \frac{1}{3} \int \frac{x^3}{1+x^2} , dx = \frac{1}{3} \int \left( x - \frac{x}{1+x^2} \right) , dx $$

Here we separated it using the identity $x^3 = x(1+x^2) - x$.

4. Integrate the separated parts

We integrate each part separately:

• The integral of $x , dx$ is $\frac{x^2}{2}$.
• The integral of $\frac{x}{1+x^2} , dx$ can be solved using the substitution $w = 1+x^2 \implies dw = 2x , dx \implies \frac{1}{2} , dw = x , dx$, giving:

$$ \int \frac{x}{1+x^2} , dx = \frac{1}{2} \ln |1+x^2| + C $$

Putting it all together, we have:

$$ \frac{x^2}{2} - \frac{1}{2} \ln |1+x^2| + C $$

5. Combine all parts to get the final result

Plugging everything back into our first equation gives:

$$ \int x^2 \arctan(x) , dx = \arctan(x) \cdot \frac{x^3}{3} - \frac{1}{3} \left( \frac{x^2}{2} - \frac{1}{2} \ln |1+x^2| \right) + C $$

Simplifying further results in:

$$ \int x^2 \arctan(x) , dx = \frac{x^3}{3} \arctan(x) - \frac{x^2}{6} + \frac{1}{6} \ln |1+x^2| + C $$