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Indefinite integralsCalculusCollegeabout 1:00

Integral of x^2 e^(x^3): animated lesson

A narrated, step-by-step indefinite integrals video lesson with captions. Final answer: e^x³/3 + 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
    ∫x2ex3 dx\int {x}^{2} {e}^{{x}^{3}}\,dx
  2. 2. Steps, part 1
    1. ∫x2ex3 dx\int {x}^{2} {e}^{{x}^{3}}\,dx
      Start with the integral
    2. ∫x2ex3 dx=13∫eu du\int {x}^{2} {e}^{{x}^{3}}\,dx = \frac{1}{3}\int {e}^{u}\,du
      Substitute u = (x³), so du = (3x²) dx
    3. ∫eu du=eu\int {e}^{u}\,du = {e}^{u}
      Exponential rule: ∫ eᵘ du = eᵘ (divide by the inner slope)
    4. ∫x2ex3 dx=ex33\int {x}^{2} {e}^{{x}^{3}}\,dx = \frac{{e}^{{x}^{3}}}{3}
      Substitute back u = (x³)
  3. 3. Steps, part 2
    1. ∫x2ex3 dx=ex33+C\int {x}^{2} {e}^{{x}^{3}}\,dx = \frac{{e}^{{x}^{3}}}{3} + C
      Simplify and add the constant of integration
  4. 4. The answer
    ex33+C\frac{{e}^{{x}^{3}}}{3} + C

    ∫ x² e^(x³) dx = e^x³/3 + 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 squared e to the power x cubed). Find the indefinite integral (antiderivative) of x squared e to the power x cubed) with respect to x.

Steps, part 1. Start with the integral. Substitute u equals (x cubed), so du equals (3x squared) dx. Exponential rule: the integral of eᵘ du equals eᵘ (divide by the inner slope). Substitute back u equals (x cubed).

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

The answer. So the answer is e to the power x cubed/3 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.

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Integral of x^2 e^(x^3): Animated Step-by-Step Lesson