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Indefinite integralsCalculusCollegeabout 0:53

Evaluate ∫ x(x+1) dx: animated lesson

A narrated, step-by-step indefinite integrals video lesson with captions. Final answer: x²/2 + x³/3 + 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
$\int x \left( x+1\right)\,dx$
2. Step by step
1. $\int x \left( x+1\right)\,dx$
  Start with the integral
2. $x \left( x+1\right) = x+{x}^{2}$
  Expand into a polynomial
3. $\int x+{x}^{2}\,dx = \frac{{x}^{2}}{2}+\frac{{x}^{3}}{3}$
  Power rule on each term: ∫ xⁿ dx = xⁿ⁺¹/(n+1)
4. $\int x \left( x+1\right)\,dx = \frac{{x}^{2}}{2}+\frac{{x}^{3}}{3} + C$
  Simplify and add the constant of integration
3. The answer
$\frac{{x}^{2}}{2}+\frac{{x}^{3}}{3} + C$
∫ x(x + 1) dx = x²/2 + x³/3 + C
4. Check
- Differentiated the answer and compared it with the integrand at 8 sample points ✓

Transcript

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

Step by step. Start with the integral. Expand into a polynomial. Power rule on each term: 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 squared/2 plus x cubed/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.

Evaluate ∫ x(x+1) dx: Animated Step-by-Step Lesson (College)