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Indefinite integralsCalculusCollegeabout 1:13

Antiderivative of sin(x)^3: animated lesson

A narrated, step-by-step indefinite integrals video lesson with captions. Final answer: cos(x)³/3 - cos(x) + 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 (0.5 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Evaluating an integral
$\int {\sin\left( x\right)}^{3}\,dx$
2. Steps, part 1
1. $\int {\sin\left( x\right)}^{3}\,dx$
  Start with the integral
2. $sin^{3}( x) = 1\sin\left( x\right) {\cos\left( x\right)}^{0}+-1\sin\left( x\right) {\cos\left( x\right)}^{2}$
  Odd power: write sin^3 u = sin u · (1 − cos² u)^1, then substitute u = cos
3. $\int 1\sin\left( x\right) {\cos\left( x\right)}^{0}\,dx = -1\int {u}^{0}\,du$
  Substitute u = cos(x), so du = (-sin(x)) dx
4. $\int {u}^{0}\,du = u$
  Power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1)
3. Steps, part 2
1. $\int 1\sin\left( x\right) {\cos\left( x\right)}^{0}\,dx = -\cos\left( x\right)$
  Substitute back u = cos(x)
2. $\int -1\sin\left( x\right) {\cos\left( x\right)}^{2}\,dx = \int {u}^{2}\,du$
  Substitute u = cos(x), so du = (-sin(x)) dx
3. $\int {\sin\left( x\right)}^{3}\,dx = \frac{{\cos\left( x\right)}^{3}}{3}-\cos\left( x\right)$
  Integrate term by term
4. $\int {\sin\left( x\right)}^{3}\,dx = \frac{{\cos\left( x\right)}^{3}}{3}-\cos\left( x\right) + C$
  Simplify and add the constant of integration
4. The answer
$\frac{{\cos\left( x\right)}^{3}}{3}-\cos\left( x\right) + C$
∫ sin(x)³ dx = cos(x)³/3 - cos(x) + C
5. Check
- Differentiated the answer and compared it with the integrand at 8 sample points ✓

Transcript

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

Steps, part 1. Start with the integral. Odd power: write sin cubed u equals sin u times (1 minus cos squared u) to the power 1, then substitute u equals cos. Substitute u equals cos(x), so du equals (negative sin(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 cos(x). Substitute u equals cos(x), so du equals (negative sin(x)) dx. Integrate term by term. Simplify and add the constant of integration.

The answer. So the answer is cos(x) cubed/3 minus cos(x) 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 sin(x)^3: Animated Step-by-Step Lesson