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Indefinite integralsCalculusCollegeabout 0:53

Integrate 6x^5: animated lesson

A narrated, step-by-step indefinite integrals video lesson with captions. Final answer: x⁶ + 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
$\int 6{x}^{5}\,dx$
2. Step by step
1. $\int 6{x}^{5}\,dx$
  Start with the integral
2. $\int {x}^{5}\,dx = \frac{{x}^{6}}{6}$
  Power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1)
3. $\int 6{x}^{5}\,dx = {x}^{6}$
  Constant multiple rule: ∫ k·f dx = k∫ f dx
4. $\int 6{x}^{5}\,dx = {x}^{6} + C$
  Simplify and add the constant of integration
3. The answer
${x}^{6} + C$
∫ 6 x⁵ dx = x⁶ + C
4. Check
- Differentiated the answer and compared it with the integrand at 8 sample points ✓

Transcript

The problem. Here is the problem: integrate 6x to the power 5. Find the indefinite integral (antiderivative) of 6 x to the power 5 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). Constant multiple rule: the integral of k times f dx equals k the integral of f dx. Simplify and add the constant of integration.

The answer. So the answer is x to the power 6 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 6x^5: Indefinite integrals Step-by-Step Lesson