Special Integrals

📖 The Two Special Integral Forms

These two integrals arise when integrating functions involving square roots with quadratic expressions. They require substitution techniques and produce inverse-trig or logarithmic results.

∫ dx / √(a² − x²) = arcsin(x/a) + C

Valid for |x| < a

∫ dx / √(a² + x²) = ln|x + √(a²+x²)| / a + C

Valid for all real x

🔤 What the Variables Mean

a

Positive constantA real number greater than zero — determines the scale of the formula

x

Variable of integrationThe independent variable

C

Constant of integrationAny real constant — added for all indefinite integrals

📝 Example 1 — Arcsin Form

Evaluate: ∫ dx / √(4 − x²)

1

Match to formula: a² = 4, so a = 2∫ dx / √(a² − x²) with a = 2

2

Apply the formula directly= arcsin(x/2) + C

✓

Answer: arcsin(x/2) + C (valid for |x| < 2)

📝 Example 2 — Logarithm Form

Evaluate: ∫ dx / √(9 + x²)

1

Match to formula: a² = 9, so a = 3∫ dx / √(a² + x²) with a = 3

2

Apply the formula= ln|x + √(9 + x²)| / 3 + C

✓

Answer: ln|x + √(9 + x²)| / 3 + C

🧮 Calculator

Compute the exact value of these special integrals at a specific x and a.

x value

a value (a > 0)

💡 Where Do These Come From?

Both integrals are derived using trigonometric substitution:

For √(a²−x²): let x = a sin θ, so dx = a cos θ dθ and the square root simplifies to a cos θ.

For √(a²+x²): let x = a tan θ, so dx = a sec²θ dθ and the square root simplifies to a sec θ.