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Indefinite integralsCalculusCollegeabout 1:13

Integrate 3x^2 + 2x: animated lesson

A narrated, step-by-step indefinite integrals video lesson with captions. Final answer: x³ + x² + 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 (0.5 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Evaluating an integral
    ∫3x2+2x dx\int 3{x}^{2}+2x\,dx
  2. 2. Steps, part 1
    1. ∫3x2+2x dx\int 3{x}^{2}+2x\,dx
      Start with the integral
    2. ∫x2 dx=x33\int {x}^{2}\,dx = \frac{{x}^{3}}{3}
      Power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1)
    3. ∫3x2 dx=x3\int 3{x}^{2}\,dx = {x}^{3}
      Constant multiple rule: ∫ k·f dx = k∫ f dx
    4. ∫x dx=x22\int x\,dx = \frac{{x}^{2}}{2}
      Power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1)
  3. 3. Steps, part 2
    1. ∫2x dx=x2\int 2x\,dx = {x}^{2}
      Constant multiple rule: ∫ k·f dx = k∫ f dx
    2. ∫3x2+2x dx=x3+x2\int 3{x}^{2}+2x\,dx = {x}^{3}+{x}^{2}
      Sum rule: integrate term by term
    3. ∫3x2+2x dx=x3+x2+C\int 3{x}^{2}+2x\,dx = {x}^{3}+{x}^{2} + C
      Simplify and add the constant of integration
  4. 4. The answer
    x3+x2+C{x}^{3}+{x}^{2} + C

    ∫ 3 x² + 2 x dx = x³ + x² + C

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

Transcript

The problem. Here is the problem: integrate 3x squared plus 2x. Find the indefinite integral (antiderivative) of 3 x squared plus 2 x with respect to x.

Steps, part 1. Start with the integral. Power rule: the integral of x to the power n dx equals x to the power n plus 1/(n plus 1). Constant multiple rule: the integral of k times f dx equals k the integral of f dx. Power rule: the integral of x to the power n dx equals x to the power n plus 1/(n plus 1).

Steps, part 2. Constant multiple rule: the integral of k times f dx equals k the integral of f dx. Sum rule: integrate term by term. Simplify and add the constant of integration.

The answer. So the answer is x cubed plus x squared 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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Integrate 3x^2 + 2x: Animated Step-by-Step Lesson (College)