Integrals of Special Forms: The Complete Formula List

Beyond the basic standard list, there is a second tier of classic integral forms that appear so often in board exams that textbooks list them separately as special forms. They involve denominators like x² + a², square roots like √(a² − x²), and their cousins. None of them yields to the power rule directly — each needs a specific trigonometric or algebraic substitution — but the results are fixed patterns you can learn once and reuse forever.

These forms matter because they are the final step of many harder problems. After a substitution like x = a tan θ, an intimidating integral collapses into one of these special forms. Recognising them quickly is what separates a smooth exam solution from a stuck one.

Key Concepts

• The x² + a² family: ∫ dx/(x² + a²) = (1/a) tan⁻¹(x/a) + C, derived with the substitution x = a tan θ.
• The √(a² − x²) family: ∫ dx/√(a² − x²) = sin⁻¹(x/a) + C, derived with x = a sin θ.
• The √(x² ± a²) family: ∫ dx/√(x² + a²) = ln|x + √(x² + a²)| + C; ∫ dx/√(x² − a²) = ln|x + √(x² − a²)| + C.
• Difference of squares: ∫ dx/(x² − a²) = (1/2a) ln|(x − a)/(x + a)| + C; ∫ dx/(a² − x²) = (1/2a) ln|(a + x)/(a − x)| + C.
• Root-times-x forms: ∫ √(a² − x²) dx = (x/2)√(a² − x²) + (a²/2) sin⁻¹(x/a) + C, and similar formulas exist for √(x² + a²) and √(x² − a²).
• Completing the square: denominators like x² + 6x + 13 must first be rewritten as (x + 3)² + 4 to reveal the special form.

Formulas

• ∫ dx/(x² + a²) = (1/a) tan⁻¹(x/a) + C
• ∫ dx/√(a² − x²) = sin⁻¹(x/a) + C
• ∫ dx/√(x² + a²) = ln|x + √(x² + a²)| + C
• ∫ dx/√(x² − a²) = ln|x + √(x² − a²)| + C
• ∫ dx/(x² − a²) = (1/2a) ln|(x − a)/(x + a)| + C
• ∫ dx/(a² − x²) = (1/2a) ln|(a + x)/(a − x)| + C

Worked Example

Evaluate ∫ dx/(x² + 6x + 13). Step 1: complete the square: x² + 6x + 13 = (x + 3)² + 4 = (x + 3)² + 2². Step 2: this matches ∫ du/(u² + a²) with u = x + 3 and a = 2. Step 3: apply the formula: (1/2) tan⁻¹((x + 3)/2) + C. So ∫ dx/(x² + 6x + 13) = (1/2) tan⁻¹((x + 3)/2) + C. Differentiating the answer with the chain rule returns 1/(x² + 6x + 13), confirming the result.

Common Misconceptions

• Confusing x² − a² with a² − x²: the first gives a log of a ratio, the second an inverse sine. Swapping them gives a completely wrong answer.
• Forgetting the 1/a factor: ∫ dx/(x² + 4) is (1/2) tan⁻¹(x/2), not tan⁻¹(x/2) — the a in the denominator of the formula matters.
• Skipping completing the square: the formula only applies to a perfect u² + a² shape; always reshape the denominator first.

Real-World Applications

These integrals compute the electric field of charge distributions, the time period of nonlinear oscillators, and arc lengths of curves. In statistics, the normal distribution's normalisation and the Cauchy distribution's cumulative function both use the x² + a² form. Signal processing uses them to analyse filter responses.

Special forms reward pattern recognition: complete the square, match the shape, apply the formula. Keep this short list beside your standard formulae, and the trickiest-looking integrals in the chapter become one-line answers.