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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.
  • 2 worked examples
  • 19 in Integrals

Worked examples

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

Answer: arcsin(x/3) + C

1 more worked examplePremium

Evaluate ∫₀^(1/2) 1/√(1-x²) dx.

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See all 19 Integrals formulas on one printable sheet - the same statements and conditions as these pages, typeset and laid out for paper.

Common questions

What is the Integral yielding arcsin?

An integral that produces the inverse sine function. Recognizing this pattern is key. It is one of the integrals formulas in the CalcRef reference.

When does the Integral yielding arcsin apply?

The Integral yielding arcsin holds under this condition: |x| < a, a > 0.

How do you use the Integral yielding arcsin?

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

Practice

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