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Indefinite integralsCalculusCollegeabout 0:47

Integral x^-3 dx: animated lesson

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

Transcript

The problem. Here is the problem: integral x to the power negative 3 dx. Find the indefinite integral (antiderivative) of x to the power negative 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 -(1 over 2) / x squared 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.

Integral x^-3 dx: Indefinite integrals Step-by-Step Lesson