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Indefinite integralsCalculusCollegeabout 0:52

Evaluate ∫ x/(x^2 + 4) dx: animated lesson

A narrated, step-by-step indefinite integrals video lesson with captions. Final answer: ln(abs(x² + 4)) / 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
    ∫xx2+4 dx\int \frac{x}{{x}^{2}+4}\,dx
  2. 2. Step by step
    1. ∫xx2+4 dx\int \frac{x}{{x}^{2}+4}\,dx
      Start with the integral
    2. ∫xx2+4 dx=ln⁡(∣x2+4∣)2\int \frac{x}{{x}^{2}+4}\,dx = \frac{\ln\left(\left|{x}^{2}+4\right|\right)}{2}
      Logarithm rule: the numerator is a constant multiple of the denominator's derivative, ∫ u′/u dx = ln|u|
    3. ∫xx2+4 dx=ln⁡(∣x2+4∣)2+C\int \frac{x}{{x}^{2}+4}\,dx = \frac{\ln\left(\left|{x}^{2}+4\right|\right)}{2} + C
      Simplify and add the constant of integration
  3. 3. The answer
    ln⁡(∣x2+4∣)2+C\frac{\ln\left(\left|{x}^{2}+4\right|\right)}{2} + C

    ∫ x / (x² + 4) dx = ln(abs(x² + 4)) / 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 x/(x squared plus 4) dx. Find the indefinite integral (antiderivative) of x / (x squared plus 4) with respect to x.

Step by step. Start with the integral. Logarithm rule: the numerator is a constant multiple of the denominator prime s derivative, the integral of u prime /u dx equals ln|u|. Simplify and add the constant of integration.

The answer. So the answer is ln(abs(x squared plus 4)) / 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.

More lessons like this

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