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Indefinite integralsCalculusCollegeabout 0:56

Integrate sin(2x)x: animated lesson

A narrated, step-by-step indefinite integrals video lesson with captions. Final answer: -(x·cos(2x) / 2) + sin(2x) / 4 + 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 (0.4 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Evaluating an integral
$\int \sin\left(2x\right) x\,dx$
2. Step by step
1. $\int \sin\left(2x\right) x\,dx$
  Start with the integral
2. $\int u\,dv = uv - \int v\,du$
  Integration by parts (tabular method)
3. $\int \sin\left(2x\right) x\,dx = -\frac{x \cos\left(2x\right)}{2}+\frac{\sin\left(2x\right)}{4}$
  Combine the alternating products
4. $\int \sin\left(2x\right) x\,dx = -\frac{x \cos\left(2x\right)}{2}+\frac{\sin\left(2x\right)}{4} + C$
  Simplify and add the constant of integration
3. The answer
$-\frac{x \cos\left(2x\right)}{2}+\frac{\sin\left(2x\right)}{4} + C$
∫ sin(2x) x dx = -(x·cos(2x) / 2) + sin(2x) / 4 + C
4. Check
- Differentiated the answer and compared it with the integrand at 8 sample points ✓

Transcript

The problem. Here is the problem: integrate sin(2x)x. Find the indefinite integral (antiderivative) of sin(2 x) x with respect to x.

Step by step. Start with the integral. Integration by parts (tabular method). Differentiate the polynomial x until it reaches 0, integrate sin(2 x) the same number of times, and combine with alternating signs. Combine the alternating products. Simplify and add the constant of integration.

The answer. So the answer is -(x times cos(2x) / 2) plus sin(2x) / 4 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 sin(2x)x: Indefinite integrals Step-by-Step Lesson