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Definite integralsCalculusCollegeabout 1:18

Definite integral of e^(-2x) from 0 to infinity: animated lesson

A narrated, step-by-step definite integrals video lesson with captions. Final answer: Converges: 1/2 ≈ 0.5.

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. 1. The problem
    Evaluating an integral
    ∫0∞e−2x dx\int_{0}^{\infty} {e}^{-2x}\,dx
  2. 2. Steps, part 1
    1. This is an improper integral
    2. ∫e−2x dx=−12e2x\int {e}^{-2x}\,dx = \frac{-\frac{1}{2}}{{e}^{2x}}
      Exponential rule: ∫ eᵘ du = eᵘ (divide by the inner slope)
    3. F(x)=−12e2xF(x) = \frac{-\frac{1}{2}}{{e}^{2x}}
      Antiderivative
    4. lim⁡x→∞F(x)=0,F(0)=−12≈−0.5\lim_{x \to \infty} F(x) = 0,\quad F(0) = -\frac{1}{2} \approx -0.5
      Evaluate the piece from 0 to inf
  3. 3. Steps, part 2
    1. ∫0∞e−2x dx=12≈0.5\int_{0}^{\infty} {e}^{-2x}\,dx = \frac{1}{2} \approx 0.5
      Add the pieces
  4. 4. The answer
    ∫0∞e−2x dx=12≈0.5\int_{0}^{\infty} {e}^{-2x}\,dx = \frac{1}{2} \approx 0.5

    Converges to 1/2 ≈ 0.5

  5. 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): 0.5

Transcript

The problem. Here is the problem: e to the power negative 2x. Evaluate the improper integral of e to the power negative 2 x) 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. Exponential rule: the integral of eᵘ du equals eᵘ (divide by the inner slope). 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: 1 over 2 approximately 0.5. 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): 0.5.

About definite integrals

Evaluate definite integrals with the fundamental theorem of calculus.

More lessons like this

Definite integral of e^(-2x) from 0 to infinity (Lesson)