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Definite integralsCalculusCollegeabout 0:43

Integrate cos(x) from 0 to pi: animated lesson

A narrated, step-by-step definite integrals video lesson with captions. Final answer: 0.

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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
$\int_{0}^{\pi} \cos\left( x\right)\,dx$
2. Step by step
1. $\int_{0}^{\pi} \cos\left( x\right)\,dx$
  Start with the definite integral
2. $\int \cos\left( x\right)\,dx = \sin\left( x\right)$
  ∫ cos x dx = sin x
3. $F(x) = \sin\left( x\right)$
  Antiderivative
4. $F(\pi) - F(0) = 0 - 0 = 0$
  Apply the fundamental theorem of calculus
3. The answer
$\int_{0}^{\pi} \cos\left( x\right)\,dx = 0$
∫ from 0 to π = 0
4. Check
- Differentiated the antiderivative and compared it with the integrand ✓
- Adaptive Simpson's rule gives 0 ✓

Transcript

The problem. Here is the problem: integrate cos(x) from 0 to pi. Evaluate the definite integral of cos(x) from x equals 0 to x equals pi.

Step by step. Start with the definite integral. the integral of cos x dx equals sin x. Antiderivative. Apply the fundamental theorem of calculus.

The answer. So the answer is 0. A definite integral gives the signed area between the curve and the x-axis: area above the axis counts as positive and area below counts as negative.

Check. Differentiated the antiderivative and compared it with the integrand, which checks out. Adaptive Simpson prime s rule gives 0, which checks out.

About definite integrals

Evaluate definite integrals with the fundamental theorem of calculus.

Integrate cos(x) from 0 to pi: Animated Step-by-Step Lesson