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Indefinite integralsCalculusCollegeabout 0:53

Integrate 1/(x^2-1): animated lesson

A narrated, step-by-step indefinite integrals video lesson with captions. Final answer: -(ln(abs(x + 1)) / 2) + ln(abs(x - 1)) / 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
    ∫1x2−1 dx\int \frac{1}{{x}^{2}-1}\,dx
  2. 2. Step by step
    1. ∫1x2−1 dx\int \frac{1}{{x}^{2}-1}\,dx
      Start with the integral
    2. ∫1x2−1 dx=−ln⁡(∣x+1∣)2+ln⁡(∣x−1∣)2\int \frac{1}{{x}^{2}-1}\,dx = -\frac{\ln\left(\left| x+1\right|\right)}{2}+\frac{\ln\left(\left| x-1\right|\right)}{2}
      Partial fractions: split 1/((x − r₁)(x − r₂)) into A/(x − r₁) + B/(x − r₂)
    3. ∫1x2−1 dx=−ln⁡(∣x+1∣)2+ln⁡(∣x−1∣)2+C\int \frac{1}{{x}^{2}-1}\,dx = -\frac{\ln\left(\left| x+1\right|\right)}{2}+\frac{\ln\left(\left| x-1\right|\right)}{2} + C
      Simplify and add the constant of integration
  3. 3. The answer
    −ln⁡(∣x+1∣)2+ln⁡(∣x−1∣)2+C-\frac{\ln\left(\left| x+1\right|\right)}{2}+\frac{\ln\left(\left| x-1\right|\right)}{2} + C

    ∫ 1 / (x² - 1) dx = -(ln(abs(x + 1)) / 2) + ln(abs(x - 1)) / 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: integrate 1/(x squared minus 1). Find the indefinite integral (antiderivative) of 1 / (x squared minus 1) with respect to x.

Step by step. Start with the integral. Partial fractions: split 1/((x minus r₁)(x minus r₂)) into A/(x minus r₁) plus B/(x minus r₂). Simplify and add the constant of integration.

The answer. So the answer is -(ln(abs(x plus 1)) / 2) plus ln(abs(x minus 1)) / 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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Integrate 1/(x^2-1): Animated Step-by-Step Lesson (College)