Mathematics

The key is to recognize that the numerator and denominator are related—the numerator contains both a piece that matches the derivative of the denominator and a leftover piece. Rewrite the fraction by dividing $x^3 + x$ by $x^2 + 1$ using polynomial long division or by splitting it strategically: $\frac{x^3 + x}{x^2 + 1} = x + \frac{0}{x^2+1}$. Once you separate it into simpler parts, one piece integrates directly and the other becomes a standard arctangent form.

The key is to recognize that the numerator and denominator are related—the numerator contains both a piece that matches the derivative of the denominator and a leftover piece. Rewrite the fraction by dividing $x^3 + x$ by $x^2 + 1$ using polynomial long division or by splitting it strategically: $\frac{x^3 + x}{x^2 + 1} = x + \frac{0}{x^2+1}$. Once you separate it into simpler parts, one piece integrates directly and the other becomes a standard arctangent form.