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Indefinite integralsCalculusCollegeabout 0:53

Integral (2x+1)^3 dx: animated lesson

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

Transcript

The problem. Here is the problem: integral (2x plus 1) cubed dx. Find the indefinite integral (antiderivative) of (2 x plus 1) cubed 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 (2 x plus 1) to the power 4 over 8 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 (2x+1)^3 dx: Animated Step-by-Step Lesson (College)