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Indefinite integralsCalculusCollegeabout 0:47

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

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

Integrate 1/(x^2+1): Animated Step-by-Step Lesson (College)