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Definite integralsCalculusCollegeabout 1:24

Integrate 1/(x^2+1) from 0 to infinity: animated lesson

A narrated, step-by-step definite integrals video lesson with captions. Final answer: Converges: π/2 ≈ 1.570796327.

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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 (0.5 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Evaluating an integral
$\int_{0}^{\infty} \frac{1}{{x}^{2}+1}\,dx$
2. Steps, part 1
1. This is an improper 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. $F(x) = \tan^{-1}\left( x\right)$
  Antiderivative
4. $\lim_{x \to \infty} F(x) = \frac{\pi}{2} \approx 1.570796327,\quad F(0) = 0$
  Evaluate the piece from 0 to inf
3. Steps, part 2
1. $\int_{0}^{\infty} \frac{1}{{x}^{2}+1}\,dx = \frac{\pi}{2} \approx 1.570796327$
  Add the pieces
4. The answer
$\int_{0}^{\infty} \frac{1}{{x}^{2}+1}\,dx = \frac{\pi}{2} \approx 1.570796327$
Converges to π/2 ≈ 1.570796327
5. Check
- Antiderivative verified by differentiation ✓
- Each limit of the antiderivative was computed and checked numerically by the limit engine ✓
- Truncated numerical estimate (stopping 10⁻⁶ from asymptotes and at ±1000 for infinite limits): 1.5698

Transcript

The problem. Here is the problem: 1/(x squared plus 1). Evaluate the improper integral of 1 / (x squared plus 1) from 0 to inf.

Steps, part 1. This is an improper integral. It is improper because a limit of integration is infinite. It is defined as a limit, and it converges only if every limit is finite. Inverse tangent form: the integral of du/(a squared plus u squared) equals (1/a) times arctan(u/a). Antiderivative. Evaluate the piece from 0 to inf.

Steps, part 2. Add the pieces. Every limit is finite, so the integral converges.

The answer. So the answer is Converges: pi/2 approximately 1.570796327. An improper integral replaces the problem point with a variable, integrates normally, and then takes a limit. A finite limit means the infinite region still has finite area.

Check. Antiderivative verified by differentiation, which checks out. Each limit of the antiderivative was computed and checked numerically by the limit engine, which checks out. Truncated numerical estimate (stopping 10 to the power minus 6 from asymptotes and at plus or minus 1000 for infinite limits): 1.5698.

About definite integrals

Evaluate definite integrals with the fundamental theorem of calculus.

Integrate 1/(x^2+1) from 0 to infinity: Step-by-Step Lesson