Verify the integration formulas in Exercises 111–114.
111. ∫ (arctan x) / x² dx = ln x - 1/2 ln(1 + x²) - arctan x / x + C

Verified step by step guidance

1. Identify the integral to verify: \(\int \frac{\arctan x}{x^{2}} \, dx\).
2. Consider using integration by parts. Let \(u = \arctan x\) and \(dv = \frac{1}{x^{2}} dx\).
3. Compute the derivatives and integrals needed: \(du = \frac{1}{1 + x^{2}} dx\) and \(v = -\frac{1}{x}\).
4. Apply the integration by parts formula: \(\int u \, dv = uv - \int v \, du\), which gives \(-\frac{\arctan x}{x} - \int \left(-\frac{1}{x} \cdot \frac{1}{1 + x^{2}}\right) dx\).
5. Simplify the remaining integral \(\int \frac{1}{x(1 + x^{2})} dx\) by using partial fraction decomposition or substitution, then combine all parts to match the given formula.

Key Concepts

Integration by Parts

Integration by parts is a technique used to integrate products of functions. It is based on the product rule for differentiation and follows the formula ∫u dv = uv - ∫v du. Choosing appropriate u and dv is crucial to simplify the integral effectively.

Derivative and Integral of Inverse Trigonometric Functions

Understanding the derivatives and integrals of inverse trigonometric functions like arctan(x) is essential. For example, d/dx [arctan x] = 1/(1 + x²). This knowledge helps in manipulating integrals involving arctan(x) and recognizing patterns during integration.

Logarithmic Integration and Simplification

Integrals often result in logarithmic expressions, especially when integrating rational functions. Recognizing when to rewrite expressions using logarithm properties, such as ln(a) - ln(b) = ln(a/b), aids in simplifying the final answer and verifying given formulas.