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Indefinite integralsCalculusCollegeabout 0:51

Evaluate ∫ 1/sqrt(4 - x^2) dx: animated lesson

A narrated, step-by-step indefinite integrals video lesson with captions. Final answer: asin(x / 2) + C.

Video with a synthetic English voice and captions. Pick another language for subtitles in that language, or open the interactive version.

Video narrated by a synthetic voice (Piper TTS, en_US-ljspeech-high voice (LJ Speech dataset, public domain)); the steps come from our maths engine and the answer was checked independently. Download MP4 (0.3 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Evaluating an integral
    ∫14−x2 dx\int \frac{1}{\sqrt{4-{x}^{2}}}\,dx
  2. 2. Step by step
    1. ∫14−x2 dx\int \frac{1}{\sqrt{4-{x}^{2}}}\,dx
      Start with the integral
    2. ∫14−x2 dx=sin⁡−1(x2)\int \frac{1}{\sqrt{4-{x}^{2}}}\,dx = \sin^{-1}\left(\frac{x}{2}\right)
      Inverse sine form: ∫ du/√(a² − u²) = arcsin(u/a)
    3. ∫14−x2 dx=sin⁡−1(x2)+C\int \frac{1}{\sqrt{4-{x}^{2}}}\,dx = \sin^{-1}\left(\frac{x}{2}\right) + C
      Simplify and add the constant of integration
  3. 3. The answer
    sin⁡−1(x2)+C\sin^{-1}\left(\frac{x}{2}\right) + C

    ∫ 1 / sqrt(4 - x²) dx = asin(x / 2) + C

  4. 4. Check
    • Differentiated the answer and compared it with the integrand at 8 sample points ✓

Transcript

The problem. Here is the problem: the integral of 1/sqrt(4 minus x squared) dx. Find the indefinite integral (antiderivative) of 1 / sqrt(4 minus x squared) with respect to x.

Step by step. Start with the integral. Inverse sine form: the integral of du/the square root of a squared minus u squared) equals arcsin(u/a). Simplify and add the constant of integration.

The answer. So the answer is asin(x / 2) plus C. An antiderivative F satisfies F prime of x equals f(x). Because the derivative of any constant is 0, every antiderivative differs by a constant C.

Check. Differentiated the answer and compared it with the integrand at 8 sample points, which checks out.

About indefinite integrals

Find antiderivatives with standard rules, substitution and integration by parts.

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Evaluate ∫ 1/sqrt(4 - x^2) dx: Animated Step-by-Step Lesson