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Integral of (sin(x/4))^3 dx

Limits of integration:

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Piecewise:

The solution

You have entered [src]
 2*p          
  /           
 |            
 |     3/x\   
 |  sin |-| dx
 |      \4/   
 |            
/             
0             
$$\int\limits_{0}^{2 p} \sin^{3}{\left(\frac{x}{4} \right)}\, dx$$
Integral(sin(x/4)^3, (x, 0, 2*p))
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 a constant times a function is the constant times the integral of the function:

          1. The integral of is when :

          So, the result is:

        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. 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 sine is negative cosine:

          So, the result is:

        Now substitute back in:

      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. 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 sine is negative cosine:

          So, the result is:

        Now substitute back in:

      The result is:

  3. Now simplify:

  4. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                 3/x\
 |                             4*cos |-|
 |    3/x\               /x\         \4/
 | sin |-| dx = C - 4*cos|-| + ---------
 |     \4/               \4/       3    
 |                                      
/                                       
$$\int \sin^{3}{\left(\frac{x}{4} \right)}\, dx = C + \frac{4 \cos^{3}{\left(\frac{x}{4} \right)}}{3} - 4 \cos{\left(\frac{x}{4} \right)}$$
The answer [src]
                    3/p\
               4*cos |-|
8        /p\         \2/
- - 4*cos|-| + ---------
3        \2/       3    
$$\frac{4 \cos^{3}{\left(\frac{p}{2} \right)}}{3} - 4 \cos{\left(\frac{p}{2} \right)} + \frac{8}{3}$$
=
=
                    3/p\
               4*cos |-|
8        /p\         \2/
- - 4*cos|-| + ---------
3        \2/       3    
$$\frac{4 \cos^{3}{\left(\frac{p}{2} \right)}}{3} - 4 \cos{\left(\frac{p}{2} \right)} + \frac{8}{3}$$
8/3 - 4*cos(p/2) + 4*cos(p/2)^3/3

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