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Indefinite integralsCalculusCollegeabout 0:48

Integral of 1/(x^2 + 9): animated lesson

A narrated, step-by-step indefinite integrals video lesson with captions. Final answer: atan(x / 3) / 3 + 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
    ∫1x2+9 dx\int \frac{1}{{x}^{2}+9}\,dx
  2. 2. Step by step
    1. ∫1x2+9 dx\int \frac{1}{{x}^{2}+9}\,dx
      Start with the integral
    2. ∫1x2+9 dx=tan⁡−1(x3)3\int \frac{1}{{x}^{2}+9}\,dx = \frac{\tan^{-1}\left(\frac{x}{3}\right)}{3}
      Inverse tangent form: ∫ du/(a² + u²) = (1/a)·arctan(u/a)
    3. ∫1x2+9 dx=tan⁡−1(x3)3+C\int \frac{1}{{x}^{2}+9}\,dx = \frac{\tan^{-1}\left(\frac{x}{3}\right)}{3} + C
      Simplify and add the constant of integration
  3. 3. The answer
    tan⁡−1(x3)3+C\frac{\tan^{-1}\left(\frac{x}{3}\right)}{3} + C

    ∫ 1 / (x² + 9) dx = atan(x / 3) / 3 + C

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

Transcript

The problem. Here is the problem: integral of 1/(x squared plus 9). Find the indefinite integral (antiderivative) of 1 / (x squared plus 9) with respect to x.

Step by step. Start with the integral. Inverse tangent form: the integral of du/(a squared plus u squared) equals (1/a) times arctan(u/a). Simplify and add the constant of integration.

The answer. So the answer is atan(x / 3) / 3 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.

More lessons like this

Integral of 1/(x^2 + 9): Animated Step-by-Step Lesson