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Integral of sinx-sin^3x/3 dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src]
  1                      
  /                      
 |                       
 |  /            3   \   
 |  |         sin (x)|   
 |  |sin(x) - -------| dx
 |  \            3   /   
 |                       
/                        
0                        
$$\int\limits_{0}^{1} \left(- \frac{\sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}\right)\, dx$$
Integral(sin(x) - sin(x)^3/3, (x, 0, 1))
Detail solution
  1. Integrate term-by-term:

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

      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:

      So, the result is:

    1. The integral of sine is negative cosine:

    The result is:

  2. Now simplify:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                              
 |                                               
 | /            3   \                        3   
 | |         sin (x)|          2*cos(x)   cos (x)
 | |sin(x) - -------| dx = C - -------- - -------
 | \            3   /             3          9   
 |                                               
/                                                
$$\int \left(- \frac{\sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}\right)\, dx = C - \frac{\cos^{3}{\left(x \right)}}{9} - \frac{2 \cos{\left(x \right)}}{3}$$
The graph
The answer [src]
                  3   
7   2*cos(1)   cos (1)
- - -------- - -------
9      3          9   
$$- \frac{2 \cos{\left(1 \right)}}{3} - \frac{\cos^{3}{\left(1 \right)}}{9} + \frac{7}{9}$$
=
=
                  3   
7   2*cos(1)   cos (1)
- - -------- - -------
9      3          9   
$$- \frac{2 \cos{\left(1 \right)}}{3} - \frac{\cos^{3}{\left(1 \right)}}{9} + \frac{7}{9}$$
7/9 - 2*cos(1)/3 - cos(1)^3/9
Numerical answer [src]
0.400050839948908
0.400050839948908

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