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Indefinite integralsCalculusCollegeabout 0:59

Integrate x sqrt(x^2 + 1): animated lesson

A narrated, step-by-step indefinite integrals video lesson with captions. Final answer: (x² + 1)^(3/2) / 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 (0.4 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Evaluating an integral
$\int x \sqrt{{x}^{2}+1}\,dx$
2. Steps, part 1
1. $\int x \sqrt{{x}^{2}+1}\,dx$
  Start with the integral
2. $\int x \sqrt{{x}^{2}+1}\,dx = \frac{1}{2}\int \sqrt{u}\,du$
  Substitute u = x² + 1, so du = (2x) dx
3. $\int \sqrt{u}\,du = \frac{{u}^{\frac{3}{2}}\cdot2}{3}$
  ∫ √u du = (2/3)u^(3/2)
4. $\int x \sqrt{{x}^{2}+1}\,dx = \frac{{\left({x}^{2}+1\right)}^{\frac{3}{2}}}{3}$
  Substitute back u = x² + 1
3. Steps, part 2
1. $\int x \sqrt{{x}^{2}+1}\,dx = \frac{{\left({x}^{2}+1\right)}^{\frac{3}{2}}}{3} + C$
  Simplify and add the constant of integration
4. The answer
$\frac{{\left({x}^{2}+1\right)}^{\frac{3}{2}}}{3} + C$
∫ x sqrt(x² + 1) dx = (x² + 1)^(3/2) / 3 + C
5. Check
- Differentiated the answer and compared it with the integrand at 8 sample points ✓

Transcript

The problem. Here is the problem: integrate x sqrt(x squared plus 1). Find the indefinite integral (antiderivative) of x sqrt(x squared plus 1) with respect to x.

Steps, part 1. Start with the integral. Substitute u equals x squared plus 1, so du equals (2x) dx. the integral of the square root of u du equals (2 over 3)u to the power 3 over 2). Substitute back u equals x squared plus 1.

Steps, part 2. Simplify and add the constant of integration.

The answer. So the answer is (x squared plus 1) to the power 3 over 2) / 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 x sqrt(x^2 + 1): Animated Step-by-Step Lesson