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Indefinite integralsCalculusCollegeabout 0:52

Integrate (x+4)^5: animated lesson

A narrated, step-by-step indefinite integrals video lesson with captions. Final answer: (x + 4)⁶/6 + 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
$\int {\left( x+4\right)}^{5}\,dx$
2. Step by step
1. $\int {\left( x+4\right)}^{5}\,dx$
  Start with the integral
2. $\int {\left( x+4\right)}^{5}\,dx = \frac{{\left( x+4\right)}^{6}}{6}$
  Linear substitution u = ax + b with the power rule: ∫ uⁿ du = uⁿ⁺¹/(n+1), divide by a
3. $\int {\left( x+4\right)}^{5}\,dx = \frac{{\left( x+4\right)}^{6}}{6} + C$
  Simplify and add the constant of integration
3. The answer
$\frac{{\left( x+4\right)}^{6}}{6} + C$
∫ (x + 4)⁵ dx = (x + 4)⁶/6 + C
4. Check
- Differentiated the answer and compared it with the integrand at 8 sample points ✓

Transcript

The problem. Here is the problem: integrate (x plus 4) to the power 5. Find the indefinite integral (antiderivative) of (x plus 4) to the power 5 with respect to x.

Step by step. Start with the integral. Linear substitution u equals ax plus b with the power rule: the integral of u to the power n du equals u to the power n plus 1/(n plus 1), divide by a. Simplify and add the constant of integration.

The answer. So the answer is (x plus 4) to the power 6 over 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 (x+4)^5: Indefinite integrals Step-by-Step Lesson