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sin^4x*cos^4xdx

Integral of sin^4x*cos^4xdx 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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 |     4       4      
 |  sin (x)*cos (x) dx
 |                    
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0                     
$$\int\limits_{0}^{1} \sin^{4}{\left(x \right)} \cos^{4}{\left(x \right)}\, dx$$
Integral(sin(x)^4*cos(x)^4, (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. 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. 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. 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. 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 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. 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. 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 is the constant times the variable of integration:

      The result is:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                  
 |                                                   
 |    4       4             sin(4*x)   sin(8*x)   3*x
 | sin (x)*cos (x) dx = C - -------- + -------- + ---
 |                            128        1024     128
/                                                    
$$\int \sin^{4}{\left(x \right)} \cos^{4}{\left(x \right)}\, dx = C + \frac{3 x}{128} - \frac{\sin{\left(4 x \right)}}{128} + \frac{\sin{\left(8 x \right)}}{1024}$$
The graph
The answer [src]
                           3          
 3    3*cos(2)*sin(2)   sin (2)*cos(2)
--- - --------------- - --------------
128         256              128      
$$- \frac{\sin^{3}{\left(2 \right)} \cos{\left(2 \right)}}{128} - \frac{3 \sin{\left(2 \right)} \cos{\left(2 \right)}}{256} + \frac{3}{128}$$
=
=
                           3          
 3    3*cos(2)*sin(2)   sin (2)*cos(2)
--- - --------------- - --------------
128         256              128      
$$- \frac{\sin^{3}{\left(2 \right)} \cos{\left(2 \right)}}{128} - \frac{3 \sin{\left(2 \right)} \cos{\left(2 \right)}}{256} + \frac{3}{128}$$
3/128 - 3*cos(2)*sin(2)/256 - sin(2)^3*cos(2)/128
Numerical answer [src]
0.0303161896573113
0.0303161896573113
The graph
Integral of sin^4x*cos^4xdx dx

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