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Indefinite integralsCalculusCollegeabout 0:46

Antiderivative of cos(x)^2: animated lesson

A narrated, step-by-step indefinite integrals video lesson with captions. Final answer: x / 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 (0.3 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Evaluating an integral
$\int {\cos\left( x\right)}^{2}\,dx$
2. Step by step
1. $\int {\cos\left( x\right)}^{2}\,dx$
  Start with the integral
2. $\int {\cos\left( x\right)}^{2}\,dx = \frac{x}{2}+\frac{\sin\left(2x\right)}{4}$
  Power-reducing identity: cos² u = (1 + cos 2u)/2
3. $\int {\cos\left( x\right)}^{2}\,dx = \frac{x}{2}+\frac{\sin\left(2x\right)}{4} + C$
  Simplify and add the constant of integration
3. The answer
$\frac{x}{2}+\frac{\sin\left(2x\right)}{4} + C$
∫ cos(x)² dx = x / 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: integral of cos(x) squared. Find the indefinite integral (antiderivative) of cos(x) squared with respect to x.

Step by step. Start with the integral. Power-reducing identity: cos squared u equals (1 plus cos 2u)/2. Simplify and add the constant of integration.

The answer. So the answer is x / 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.

Antiderivative of cos(x)^2: Animated Step-by-Step Lesson