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Indefinite integralsCalculusCollegeabout 0:51

Evaluate ∫ 2x/(x^2+1) dx: animated lesson

A narrated, step-by-step indefinite integrals video lesson with captions. Final answer: ln(abs(x² + 1)) + C.

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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. The problem
Evaluating an integral
$\int \frac{2x}{{x}^{2}+1}\,dx$
2. Step by step
1. $\int \frac{2x}{{x}^{2}+1}\,dx$
  Start with the integral
2. $\int \frac{2x}{{x}^{2}+1}\,dx = \ln\left(\left|{x}^{2}+1\right|\right)$
  Logarithm rule: the numerator is a constant multiple of the denominator's derivative, ∫ u′/u dx = ln|u|
3. $\int \frac{2x}{{x}^{2}+1}\,dx = \ln\left(\left|{x}^{2}+1\right|\right) + C$
  Simplify and add the constant of integration
3. The answer
$\ln\left(\left|{x}^{2}+1\right|\right) + C$
∫ 2 x / (x² + 1) dx = ln(abs(x² + 1)) + C
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 2x/(x squared plus 1) dx. Find the indefinite integral (antiderivative) of 2 x / (x squared plus 1) 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 1)) 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.

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