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Indefinite integralsCalculusCollegeabout 0:46

Evaluate ∫ sin(2x) dx: animated lesson

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

Transcript

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

Step by step. Start with the integral. Linear substitution u equals 2 x: the integral of sin u du equals negative cos u, then divide by 2. Simplify and add the constant of integration.

The answer. So the answer is -(cos(2 x) / 2) 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.

Evaluate ∫ sin(2x) dx: Animated Step-by-Step Lesson