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Integral of sin^8(x/4) dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src]
 2*p          
  /           
 |            
 |     8/x\   
 |  sin |-| dx
 |      \4/   
 |            
/             
0             
$$\int\limits_{0}^{2 p} \sin^{8}{\left(\frac{x}{4} \right)}\, dx$$
Integral(sin(x/4)^8, (x, 0, 2*p))
Detail solution
  1. Rewrite the integrand:

  2. There are multiple ways to do this integral.

    Method #1

    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. Rewrite the integrand:

        2. Rewrite the integrand:

        3. 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. 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 cosine is sine:

                    So, the result is:

                  Now substitute back in:

                So, the result is:

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

              The result is:

            So, the result is:

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

            1. The integral of cosine is sine:

            So, the result is:

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

          The result is:

        So, the result is:

      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 a constant is the constant times the variable of integration:

              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:

              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 cosine is sine:

                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 cosine is sine:

                So, the result is:

              Now substitute back in:

            The result is:

        So, the result is:

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

        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. The integral of cosine is sine:

            So, the result is:

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

          The result is:

        So, the result is:

      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 cosine is sine:

            So, the result is:

          Now substitute back in:

        So, the result is:

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

      The result is:

    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. Rewrite the integrand:

        2. Rewrite the integrand:

        3. 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. 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 cosine is sine:

                    So, the result is:

                  Now substitute back in:

                So, the result is:

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

              The result is:

            So, the result is:

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

            1. The integral of cosine is sine:

            So, the result is:

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

          The result is:

        So, the result is:

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

        1. Rewrite the integrand:

        2. Let .

          Then let and substitute :

          1. Integrate term-by-term:

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

            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:

            The result is:

          Now substitute back in:

        So, the result is:

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

        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. The integral of cosine is sine:

            So, the result is:

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

          The result is:

        So, the result is:

      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 cosine is sine:

            So, the result is:

          Now substitute back in:

        So, the result is:

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

      The result is:

  3. Now simplify:

  4. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                             3/x\                             
 |                           sin |-|                             
 |    8/x\             /x\       \2/   sin(2*x)   7*sin(x)   35*x
 | sin |-| dx = C - sin|-| + ------- + -------- + -------- + ----
 |     \4/             \2/      6        256         32      128 
 |                                                               
/                                                                
$$\int \sin^{8}{\left(\frac{x}{4} \right)}\, dx = C + \frac{35 x}{128} + \frac{\sin^{3}{\left(\frac{x}{2} \right)}}{6} - \sin{\left(\frac{x}{2} \right)} + \frac{7 \sin{\left(x \right)}}{32} + \frac{\sin{\left(2 x \right)}}{256}$$
The answer [src]
             /p\    /p\         3/p\    /p\        5/p\    /p\      7/p\    /p\
       35*cos|-|*sin|-|   35*sin |-|*cos|-|   7*sin |-|*cos|-|   sin |-|*cos|-|
35*p         \2/    \2/          \2/    \2/         \2/    \2/       \2/    \2/
---- - ---------------- - ----------------- - ---------------- - --------------
 64           32                  48                 12                2       
$$\frac{35 p}{64} - \frac{\sin^{7}{\left(\frac{p}{2} \right)} \cos{\left(\frac{p}{2} \right)}}{2} - \frac{7 \sin^{5}{\left(\frac{p}{2} \right)} \cos{\left(\frac{p}{2} \right)}}{12} - \frac{35 \sin^{3}{\left(\frac{p}{2} \right)} \cos{\left(\frac{p}{2} \right)}}{48} - \frac{35 \sin{\left(\frac{p}{2} \right)} \cos{\left(\frac{p}{2} \right)}}{32}$$
=
=
             /p\    /p\         3/p\    /p\        5/p\    /p\      7/p\    /p\
       35*cos|-|*sin|-|   35*sin |-|*cos|-|   7*sin |-|*cos|-|   sin |-|*cos|-|
35*p         \2/    \2/          \2/    \2/         \2/    \2/       \2/    \2/
---- - ---------------- - ----------------- - ---------------- - --------------
 64           32                  48                 12                2       
$$\frac{35 p}{64} - \frac{\sin^{7}{\left(\frac{p}{2} \right)} \cos{\left(\frac{p}{2} \right)}}{2} - \frac{7 \sin^{5}{\left(\frac{p}{2} \right)} \cos{\left(\frac{p}{2} \right)}}{12} - \frac{35 \sin^{3}{\left(\frac{p}{2} \right)} \cos{\left(\frac{p}{2} \right)}}{48} - \frac{35 \sin{\left(\frac{p}{2} \right)} \cos{\left(\frac{p}{2} \right)}}{32}$$
35*p/64 - 35*cos(p/2)*sin(p/2)/32 - 35*sin(p/2)^3*cos(p/2)/48 - 7*sin(p/2)^5*cos(p/2)/12 - sin(p/2)^7*cos(p/2)/2

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