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Definite integralsCalculusCollegeabout 0:53

Integral of 1/x from 1 to e: animated lesson

A narrated, step-by-step definite integrals video lesson with captions. Final answer: 1.

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
    ∫1e1x dx\int_{1}^{e} \frac{1}{x}\,dx
  2. 2. Steps, part 1
    1. ∫1e1x dx\int_{1}^{e} \frac{1}{x}\,dx
      Start with the definite integral
    2. ∫x−1 dx=ln⁡(∣x∣)\int {x}^{-1}\,dx = \ln\left(\left| x\right|\right)
      Reciprocal rule: ∫ 1/x dx = ln|x|
    3. ∫1x−1 dx=ln⁡(∣x∣)\int 1{x}^{-1}\,dx = \ln\left(\left| x\right|\right)
      Constant multiple rule: ∫ k·f dx = k∫ f dx
    4. F(x)=ln⁡(∣x∣)F(x) = \ln\left(\left| x\right|\right)
      Antiderivative
  3. 3. Steps, part 2
    1. F(e)−F(1)=1−0=1F( e) - F(1) = 1 - 0 = 1
      Apply the fundamental theorem of calculus
  4. 4. The answer
    ∫1e1x dx=1\int_{1}^{e} \frac{1}{x}\,dx = 1

    ∫ from 1 to e = 1

  5. 5. Check
    • Differentiated the antiderivative and compared it with the integrand ✓
    • Adaptive Simpson's rule gives 1 ✓

Transcript

The problem. Here is the problem: integral of 1/x from 1 to e. Evaluate the definite integral of 1 / x from x equals 1 to x equals e.

Steps, part 1. Start with the definite integral. Reciprocal rule: the integral of 1/x dx equals ln|x|. Constant multiple rule: the integral of k times f dx equals k the integral of f dx. Antiderivative.

Steps, part 2. Apply the fundamental theorem of calculus.

The answer. So the answer is 1. A definite integral gives the signed area between the curve and the x-axis: area above the axis counts as positive and area below counts as negative.

Check. Differentiated the antiderivative and compared it with the integrand, which checks out. Adaptive Simpson prime s rule gives 1, which checks out.

About definite integrals

Evaluate definite integrals with the fundamental theorem of calculus.

More lessons like this

Integral of 1/x from 1 to e: Animated Step-by-Step Lesson