Skip to content
MathSolver Pro
  • Open the math solver
  • Ask the AI tutor
  • Find an AP course
  • Create a quiz
  • Open the calculator
  • Graphing Calculator
  • Browse all calculators
  • Read math guides
↑ ↓Enter to open · Esc to close
Solve a problem
Definite integralsCalculusCollegeabout 0:57

Integrate x e^x from 0 to 1: 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 (0.4 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Evaluating an integral
    ∫01xex dx\int_{0}^{1} x {e}^{x}\,dx
  2. 2. Steps, part 1
    1. ∫01xex dx\int_{0}^{1} x {e}^{x}\,dx
      Start with the definite integral
    2. ∫u dv=uv−∫v du\int u\,dv = uv - \int v\,du
      Integration by parts (tabular method)
    3. ∫xex dx=ex(x−1)\int x {e}^{x}\,dx = {e}^{x} \left( x-1\right)
      Combine the alternating products
    4. F(x)=ex(x−1)F(x) = {e}^{x} \left( x-1\right)
      Antiderivative
  3. 3. Steps, part 2
    1. F(1)−F(0)=0−(−1)=1F(1) - F(0) = 0 - \left(-1\right) = 1
      Apply the fundamental theorem of calculus
  4. 4. The answer
    ∫01xex dx=1\int_{0}^{1} x {e}^{x}\,dx = 1

    ∫ from 0 to 1 = 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: integrate x e to the power x from 0 to 1. Evaluate the definite integral of x e to the power x from x equals 0 to x equals 1.

Steps, part 1. Start with the definite integral. Integration by parts (tabular method). Differentiate the polynomial x until it reaches 0, integrate e to the power x the same number of times, and combine with alternating signs. Combine the alternating products. 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

Integrate x e^x from 0 to 1: Animated Step-by-Step Lesson