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Indefinite integralsCalculusCollegeabout 0:48

Antiderivative of x^(2/3): animated lesson

A narrated, step-by-step indefinite integrals video lesson with captions. Final answer: x^(5/3)·3/5 + C.

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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 (0.3 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Evaluating an integral
$\int {x}^{\frac{2}{3}}\,dx$
2. Step by step
1. $\int {x}^{\frac{2}{3}}\,dx$
  Start with the integral
2. $\int {x}^{\frac{2}{3}}\,dx = \frac{{x}^{\frac{5}{3}}\cdot3}{5}$
  Power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1)
3. $\int {x}^{\frac{2}{3}}\,dx = \frac{{x}^{\frac{5}{3}}\cdot3}{5} + C$
  Simplify and add the constant of integration
3. The answer
$\frac{{x}^{\frac{5}{3}}\cdot3}{5} + C$
∫ x^(2/3) dx = x^(5/3)·3/5 + C
4. Check
- Differentiated the answer and compared it with the integrand at 8 sample points ✓

Transcript

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

Step by step. 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). Simplify and add the constant of integration.

The answer. So the answer is x to the power 5 over 3) times 3 over 5 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.

Antiderivative of x^(2/3): Animated Step-by-Step Lesson