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Definite integralsCalculusCollegeabout 1:17

Integrate x^2 - 4x + 3 from 0 to 3: animated lesson

A narrated, step-by-step definite integrals video lesson with captions. Final answer: 0.

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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. The problem
Evaluating an integral
$\int_{0}^{3} {x}^{2}-4x+3\,dx$
2. Steps, part 1
1. $\int_{0}^{3} {x}^{2}-4x+3\,dx$
  Start with the definite integral
2. $\int {x}^{2}\,dx = \frac{{x}^{3}}{3}$
  Power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1)
3. $\int x\,dx = \frac{{x}^{2}}{2}$
  Power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1)
4. $\int 4x\,dx = 2{x}^{2}$
  Constant multiple rule: ∫ k·f dx = k∫ f dx
3. Steps, part 2
1. $\int {x}^{2}-4x\,dx = \frac{{x}^{3}}{3}-2{x}^{2}$
  Sum rule: integrate term by term
2. $\int 3\,dx = 3x$
  Constant rule: ∫ k dx = kx
3. $\int {x}^{2}-4x+3\,dx = \frac{{x}^{3}}{3}+3x-2{x}^{2}$
  Sum rule: integrate term by term
4. $F(x) = \frac{{x}^{3}}{3}+3x-2{x}^{2}$
  Antiderivative
4. Steps, part 3
1. $F(3) - F(0) = 0 - 0 = 0$
  Apply the fundamental theorem of calculus
5. The answer
$\int_{0}^{3} {x}^{2}-4x+3\,dx = 0$
∫ from 0 to 3 = 0
6. Check
- Differentiated the antiderivative and compared it with the integrand ✓
- Adaptive Simpson's rule gives 0 ✓

Transcript

The problem. Here is the problem: integrate x squared minus 4x plus 3 from 0 to 3. Evaluate the definite integral of x squared minus 4 x plus 3 from x equals 0 to x equals 3.

Steps, part 1. Start with the definite 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. Antiderivative.

Steps, part 3. Apply the fundamental theorem of calculus.

The answer. So the answer is 0. 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 0, which checks out.

About definite integrals

Evaluate definite integrals with the fundamental theorem of calculus.

Integrate x^2 - 4x + 3 from 0 to 3: Step-by-Step Lesson