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sin^3*x

Integral of sin^3*x dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src]
 pi           
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 2            
  /           
 |            
 |     3      
 |  sin (x) dx
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0             
$$\int\limits_{0}^{\frac{\pi}{2}} \sin^{3}{\left(x \right)}\, dx$$
Integral(sin(x)^3, (x, 0, pi/2))
Detail solution
  1. Rewrite the integrand:

  2. There are multiple ways to do this integral.

    Method #1

    1. Let .

      Then let and substitute :

      1. Integrate term-by-term:

        1. The integral of is when :

        1. The integral of a constant is the constant times the variable of integration:

        The result is:

      Now substitute back in:

    Method #2

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      1. The integral of a constant times a function is the constant times the integral of the function:

        1. Let .

          Then let and substitute :

          1. The integral of a constant times a function is the constant times the integral of the function:

            1. The integral of is when :

            So, the result is:

          Now substitute back in:

        So, the result is:

      1. The integral of sine is negative cosine:

      The result is:

    Method #3

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      1. The integral of a constant times a function is the constant times the integral of the function:

        1. Let .

          Then let and substitute :

          1. The integral of a constant times a function is the constant times the integral of the function:

            1. The integral of is when :

            So, the result is:

          Now substitute back in:

        So, the result is:

      1. The integral of sine is negative cosine:

      The result is:

  3. Now simplify:

  4. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                 
 |                              3   
 |    3                      cos (x)
 | sin (x) dx = C - cos(x) + -------
 |                              3   
/                                   
$$\int \sin^{3}{\left(x \right)}\, dx = C + \frac{\cos^{3}{\left(x \right)}}{3} - \cos{\left(x \right)}$$
The graph
The answer [src]
2/3
$$\frac{2}{3}$$
=
=
2/3
$$\frac{2}{3}$$
2/3
Numerical answer [src]
0.666666666666667
0.666666666666667
The graph
Integral of sin^3*x dx

    Use the examples entering the upper and lower limits of integration.