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Indefinite integralsCalculusCollegeabout 1:15

Integral of x^3 - 2x + 1: animated lesson

A narrated, step-by-step indefinite integrals video lesson with captions. Final answer: x⁴/4 + x - 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
    ∫x3−2x+1 dx\int {x}^{3}-2x+1\,dx
  2. 2. Steps, part 1
    1. ∫x3−2x+1 dx\int {x}^{3}-2x+1\,dx
      Start with the integral
    2. ∫x3 dx=x44\int {x}^{3}\,dx = \frac{{x}^{4}}{4}
      Power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1)
    3. ∫x dx=x22\int x\,dx = \frac{{x}^{2}}{2}
      Power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1)
    4. ∫2x dx=x2\int 2x\,dx = {x}^{2}
      Constant multiple rule: ∫ k·f dx = k∫ f dx
  3. 3. Steps, part 2
    1. ∫x3−2x dx=x44−x2\int {x}^{3}-2x\,dx = \frac{{x}^{4}}{4}-{x}^{2}
      Sum rule: integrate term by term
    2. ∫1 dx=x\int 1\,dx = x
      Constant rule: ∫ k dx = kx
    3. ∫x3−2x+1 dx=x44+x−x2\int {x}^{3}-2x+1\,dx = \frac{{x}^{4}}{4}+ x-{x}^{2}
      Sum rule: integrate term by term
    4. ∫x3−2x+1 dx=x44+x−x2+C\int {x}^{3}-2x+1\,dx = \frac{{x}^{4}}{4}+ x-{x}^{2} + C
      Simplify and add the constant of integration
  4. 4. The answer
    x44+x−x2+C\frac{{x}^{4}}{4}+ x-{x}^{2} + C

    ∫ x³ - 2 x + 1 dx = x⁴/4 + x - 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: integral of x cubed minus 2x plus 1. Find the indefinite integral (antiderivative) of x cubed minus 2 x plus 1 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). 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.

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

The answer. So the answer is x to the power 4 over 4 plus x minus 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.

More lessons like this

Integral of x^3 - 2x + 1: Animated Step-by-Step Lesson