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Indefinite integralsCalculusCollegeabout 1:12

Integrate cos(x)^3: animated lesson

A narrated, step-by-step indefinite integrals video lesson with captions. Final answer: sin(x) - sin(x)³/3 + C.

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 (0.5 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Evaluating an integral
    ∫cos⁡(x)3 dx\int {\cos\left( x\right)}^{3}\,dx
  2. 2. Steps, part 1
    1. ∫cos⁡(x)3 dx\int {\cos\left( x\right)}^{3}\,dx
      Start with the integral
    2. cos3(x)=1cos⁡(x)sin⁡(x)0+−1cos⁡(x)sin⁡(x)2cos^{3}( x) = 1\cos\left( x\right) {\sin\left( x\right)}^{0}+-1\cos\left( x\right) {\sin\left( x\right)}^{2}
      Odd power: write cos^3 u = cos u · (1 − sin² u)^1, then substitute u = sin
    3. ∫1cos⁡(x)sin⁡(x)0 dx=∫u0 du\int 1\cos\left( x\right) {\sin\left( x\right)}^{0}\,dx = \int {u}^{0}\,du
      Substitute u = sin(x), so du = (cos(x)) dx
    4. ∫u0 du=u\int {u}^{0}\,du = u
      Power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1)
  3. 3. Steps, part 2
    1. ∫1cos⁡(x)sin⁡(x)0 dx=sin⁡(x)\int 1\cos\left( x\right) {\sin\left( x\right)}^{0}\,dx = \sin\left( x\right)
      Substitute back u = sin(x)
    2. ∫−1cos⁡(x)sin⁡(x)2 dx=−1∫u2 du\int -1\cos\left( x\right) {\sin\left( x\right)}^{2}\,dx = -1\int {u}^{2}\,du
      Substitute u = sin(x), so du = (cos(x)) dx
    3. ∫cos⁡(x)3 dx=sin⁡(x)−sin⁡(x)33\int {\cos\left( x\right)}^{3}\,dx = \sin\left( x\right)-\frac{{\sin\left( x\right)}^{3}}{3}
      Integrate term by term
    4. ∫cos⁡(x)3 dx=sin⁡(x)−sin⁡(x)33+C\int {\cos\left( x\right)}^{3}\,dx = \sin\left( x\right)-\frac{{\sin\left( x\right)}^{3}}{3} + C
      Simplify and add the constant of integration
  4. 4. The answer
    sin⁡(x)−sin⁡(x)33+C\sin\left( x\right)-\frac{{\sin\left( x\right)}^{3}}{3} + C

    ∫ cos(x)³ dx = sin(x) - sin(x)³/3 + C

  5. 5. Check
    • Differentiated the answer and compared it with the integrand at 8 sample points ✓

Transcript

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

Steps, part 1. Start with the integral. Odd power: write cos cubed u equals cos u times (1 minus sin squared u) to the power 1, then substitute u equals sin. Substitute u equals sin(x), so du equals (cos(x)) dx. Power rule: the integral of x to the power n dx equals x to the power n plus 1/(n plus 1).

Steps, part 2. Substitute back u equals sin(x). Substitute u equals sin(x), so du equals (cos(x)) dx. Integrate term by term. Simplify and add the constant of integration.

The answer. So the answer is sin(x) minus sin(x) cubed/3 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.

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Integrate cos(x)^3: Indefinite integrals Step-by-Step Lesson