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
Indefinite integralsCalculusCollegeabout 0:58

Integral of x e^(x^2): animated lesson

A narrated, step-by-step indefinite integrals video lesson with captions. Final answer: e^x²/2 + C.

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
    ∫xex2 dx\int x {e}^{{x}^{2}}\,dx
  2. 2. Steps, part 1
    1. ∫xex2 dx\int x {e}^{{x}^{2}}\,dx
      Start with the integral
    2. ∫xex2 dx=12∫eu du\int x {e}^{{x}^{2}}\,dx = \frac{1}{2}\int {e}^{u}\,du
      Substitute u = (x²), so du = (2x) dx
    3. ∫eu du=eu\int {e}^{u}\,du = {e}^{u}
      Exponential rule: ∫ eᵘ du = eᵘ (divide by the inner slope)
    4. ∫xex2 dx=ex22\int x {e}^{{x}^{2}}\,dx = \frac{{e}^{{x}^{2}}}{2}
      Substitute back u = (x²)
  3. 3. Steps, part 2
    1. ∫xex2 dx=ex22+C\int x {e}^{{x}^{2}}\,dx = \frac{{e}^{{x}^{2}}}{2} + C
      Simplify and add the constant of integration
  4. 4. The answer
    ex22+C\frac{{e}^{{x}^{2}}}{2} + C

    ∫ x e^(x²) dx = e^x²/2 + C

  5. 5. Check
    • Differentiated the answer and compared it with the integrand at 8 sample points ✓

Transcript

The problem. Here is the problem: integral of x e to the power x squared). Find the indefinite integral (antiderivative) of x e to the power x squared) with respect to x.

Steps, part 1. Start with the integral. Substitute u equals (x squared), so du equals (2x) dx. Exponential rule: the integral of eᵘ du equals eᵘ (divide by the inner slope). Substitute back u equals (x squared).

Steps, part 2. Simplify and add the constant of integration.

The answer. So the answer is e to the power x squared/2 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.

More lessons like this

Integral of x e^(x^2): Animated Step-by-Step Lesson