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Indefinite integralsCalculusCollegeabout 0:46

Integrate sqrt(x): animated lesson

A narrated, step-by-step indefinite integrals video lesson with captions. Final answer: x^(3/2)·2/3 + 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
$\int \sqrt{x}\,dx$
2. Step by step
1. $\int \sqrt{x}\,dx$
  Start with the integral
2. $\int \sqrt{x}\,dx = \frac{{x}^{\frac{3}{2}}\cdot2}{3}$
  ∫ √x dx = (2/3)x^(3/2)
3. $\int \sqrt{x}\,dx = \frac{{x}^{\frac{3}{2}}\cdot2}{3} + C$
  Simplify and add the constant of integration
3. The answer
$\frac{{x}^{\frac{3}{2}}\cdot2}{3} + C$
∫ sqrt(x) dx = x^(3/2)·2/3 + C
4. Check
- Differentiated the answer and compared it with the integrand at 8 sample points ✓

Transcript

The problem. Here is the problem: integrate sqrt(x). Find the indefinite integral (antiderivative) of sqrt(x) with respect to x.

Step by step. Start with the integral. the integral of the square root of x dx equals (2 over 3)x to the power 3 over 2). Simplify and add the constant of integration.

The answer. So the answer is x to the power 3 over 2) times 2 over 3 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.

Integrate sqrt(x): Indefinite integrals Step-by-Step Lesson