Calculus — Integrals⏱ 6 min read

Special Integrals

Q: Where do these formulas come from?

Both come from trigonometric substitution: x = a sin θ for the arcsin form, and x = a tan θ for the logarithmic form — each substitution simplifies the square root to a single trig function.

Q: How are these related to inverse trig derivatives?

The arcsin form is the direct reverse of d/dx(arcsin x) = 1/√(1-x²), generalized with the constant a.

∫ dx/√(a²−x²) = arcsin(x/a) + C

What is the Special Integrals?

These two integrals arise when integrating expressions involving square roots of quadratic terms, and are derived using trigonometric substitution. For √(a²−x²), substituting x = a sin θ simplifies the square root to a cos θ, producing the arcsin result. For √(a²+x²), substituting x = a tan θ simplifies it to a sec θ, producing the logarithmic result.

The arcsin form is only valid for |x| < a; the logarithmic form holds for all real x.

What Each Variable Means

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

Variable of integrationThe independent variable.

Constant of integrationAdded for any indefinite integral.

When to Use It

Integrating expressions with a square root of a difference or sum of squares in the denominator
Recognizing the reverse of the inverse trig derivative formulas
As a component of more complex integration problems solved via trigonometric substitution

Step-by-Step Examples

Example 1: Arcsin form

Problem: Evaluate ∫ dx / √(4 − x²)

Match to the formula

Here a² = 4, so a = 2.

∫ dx / √(a² − x²) with a = 2

Apply the formula directly

Substitute a = 2.

= arcsin(x/2) + C

✓

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

Example 2: Logarithm form

Problem: Evaluate ∫ dx / √(9 + x²)

Match to the formula

Here a² = 9, so a = 3.

∫ dx / √(a² + x²) with a = 3

Apply the formula

Substitute a = 3.

= ln|x + √(9 + x²)| / 3 + C

✓

Answer: ln|x + √(9 + x²)| / 3 + C

Interactive Calculator

x valuea value (a > 0)

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Common Mistakes

Mistake: Using the arcsin form when |x| ≥ a.
Fix: The arcsin form ∫dx/√(a²-x²) = arcsin(x/a) + C is only valid for |x| < a — outside that range, the expression under the square root would be negative.
Mistake: Mixing up the plus and minus versions.
Fix: √(a²−x²) (a difference) gives the arcsin form; √(a²+x²) (a sum) gives the logarithm form — the sign inside the root determines which formula applies.

Practice Questions

Evaluate ∫ dx / √(16 − x²).
Hint: a² = 16, so a = 4.
Evaluate ∫ dx / √(25 + x²).

Frequently Asked Questions

Where do these formulas come from?

Both come from trigonometric substitution: x = a sin θ for the arcsin form, and x = a tan θ for the logarithmic form — each substitution simplifies the square root to a single trig function.

How are these related to inverse trig derivatives?

The arcsin form is the direct reverse of d/dx(arcsin x) = 1/√(1-x²), generalized with the constant a.