CalcRef

Integrals · Special Integrals

Integral yielding arcsin

∫1a2−x2 dx=arcsin⁡(xa)+C\int \frac{1}{\sqrt{a^2 - x^2}} \, dx = \arcsin\left(\frac{x}{a}\right) + C

An integral that produces the inverse sine function. Recognizing this pattern is key.

Conditions. |x| < a, a > 0.

Worked examples

Find ∫ 1/√(9-x²) dx.
  1. a = 3. arcsin(x/3) + C

Answer: arcsin(x/3) + C

Evaluate ∫₀^(1/2) 1/√(1-x²) dx.
  1. [arcsin x]₀^(1/2) = arcsin(1/2) - arcsin(0) = π/6 - 0

Answer: π/6

Related formulas

Integral yielding arctan

Practice this and 135 more formulas in the CalcRef workspace — quizzes, reference tables, a 16-category unit converter, and an expression evaluator.