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Definite integralsCalculusCollegeabout 1:03

Definite integral of 2x + 3 from 1 to 4: animated lesson

A narrated, step-by-step definite integrals video lesson with captions. Final answer: 24.

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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 (0.4 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Evaluating an integral
$\int_{1}^{4} 2x+3\,dx$
2. Steps, part 1
1. $\int_{1}^{4} 2x+3\,dx$
  Start with the definite integral
2. $\int x\,dx = \frac{{x}^{2}}{2}$
  Power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1)
3. $\int 2x\,dx = {x}^{2}$
  Constant multiple rule: ∫ k·f dx = k∫ f dx
4. $\int 3\,dx = 3x$
  Constant rule: ∫ k dx = kx
3. Steps, part 2
1. $\int 2x+3\,dx = {x}^{2}+3x$
  Sum rule: integrate term by term
2. $F(x) = {x}^{2}+3x$
  Antiderivative
3. $F(4) - F(1) = 28 - 4 = 24$
  Apply the fundamental theorem of calculus
4. The answer
$\int_{1}^{4} 2x+3\,dx = 24$
∫ from 1 to 4 = 24
5. Check
- Differentiated the antiderivative and compared it with the integrand ✓
- Adaptive Simpson's rule gives 24 ✓

Transcript

The problem. Here is the problem: integral of 2x plus 3 from 1 to 4. Evaluate the definite integral of 2 x plus 3 from x equals 1 to x equals 4.

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). Constant multiple rule: the integral of k times f dx equals k the integral of f dx. Constant rule: the integral of k dx equals kx.

Steps, part 2. Sum rule: integrate term by term. Antiderivative. Apply the fundamental theorem of calculus.

The answer. So the answer is 24. 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 24, which checks out.

About definite integrals

Evaluate definite integrals with the fundamental theorem of calculus.

Definite integral of 2x + 3 from 1 to 4: Step-by-Step Lesson