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sin(x)^10

Integral of sin(x)^10 dx

Limits of integration:

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The graph:

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Piecewise:

The solution

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  1            
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 |             
 |     10      
 |  sin  (x) dx
 |             
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0              
$$\int\limits_{0}^{1} \sin^{10}{\left(x \right)}\, dx$$
Integral(sin(x)^10, (x, 0, 1))
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. 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. Let .

                  Then let and substitute :

                  1. The integral of is when :

                  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. Let .

                  Then let and substitute :

                  1. The integral of is when :

                  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. The integral of cosine is sine:

                So, the result is:

              The result is:

            Now substitute back in:

          Method #2

          1. Rewrite the integrand:

          2. Integrate term-by-term:

            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:

            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. 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:

            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. 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. 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. 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. 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. 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. 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. Let .

                Then let and substitute :

                1. The integral of is when :

                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. Let .

                Then let and substitute :

                1. The integral of is when :

                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. The integral of cosine is sine:

              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. 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. 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. 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. 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. 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. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                                                    
 |                                 5           3                                       
 |    10             sin(2*x)   sin (2*x)   sin (2*x)   5*sin(8*x)   15*sin(4*x)   63*x
 | sin  (x) dx = C - -------- - --------- + --------- + ---------- + ----------- + ----
 |                      4          320          16         2048          256       256 
/                                                                                      
$${{{{5\,\left({{{{\sin \left(8\,x\right)}\over{2}}+4\,x}\over{8}}+{{ \sin \left(4\,x\right)}\over{2}}+x\right)}\over{64}}+{{5\,\left({{ \sin \left(4\,x\right)}\over{2}}+2\,x\right)}\over{32}}-{{{{\sin ^5 \left(2\,x\right)}\over{5}}-{{2\,\sin ^3\left(2\,x\right)}\over{3}}+ \sin \left(2\,x\right)}\over{32}}-{{5\,\left(\sin \left(2\,x\right)- {{\sin ^3\left(2\,x\right)}\over{3}}\right)}\over{16}}-{{5\,\sin \left(2\,x\right)}\over{32}}+{{x}\over{16}}}\over{2}}$$
The graph
The answer [src]
                               3                   5                  7                9          
 63   63*cos(1)*sin(1)   21*sin (1)*cos(1)   21*sin (1)*cos(1)   9*sin (1)*cos(1)   sin (1)*cos(1)
--- - ---------------- - ----------------- - ----------------- - ---------------- - --------------
256         256                 128                 160                 80                10      
$${{25\,\sin 8+600\,\sin 4-32\,\sin ^52+640\,\sin ^32-2560\,\sin 2+ 2520}\over{10240}}$$
=
=
                               3                   5                  7                9          
 63   63*cos(1)*sin(1)   21*sin (1)*cos(1)   21*sin (1)*cos(1)   9*sin (1)*cos(1)   sin (1)*cos(1)
--- - ---------------- - ----------------- - ----------------- - ---------------- - --------------
256         256                 128                 160                 80                10      
$$- \frac{63 \sin{\left(1 \right)} \cos{\left(1 \right)}}{256} - \frac{21 \sin^{3}{\left(1 \right)} \cos{\left(1 \right)}}{128} - \frac{21 \sin^{5}{\left(1 \right)} \cos{\left(1 \right)}}{160} - \frac{9 \sin^{7}{\left(1 \right)} \cos{\left(1 \right)}}{80} - \frac{\sin^{9}{\left(1 \right)} \cos{\left(1 \right)}}{10} + \frac{63}{256}$$
Numerical answer [src]
0.0218875224217298
0.0218875224217298
The graph
Integral of sin(x)^10 dx

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