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Integral of (sin^4)x×(cos^4)x dx

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The solution

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 |  sin (x)*x*cos (x)*x dx
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$$\int\limits_{0}^{1} x x \sin^{4}{\left(x \right)} \cos^{4}{\left(x \right)}\, dx$$
Integral(sin(x)^4*x*cos(x)^4*x, (x, 0, 1))
Detail solution
  1. Use integration by parts:

    Let and let .

    Then .

    To find :

    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:

    Now evaluate the sub-integral.

  2. Use integration by parts:

    Let and let .

    Then .

    To find :

    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 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 sine is negative cosine:

            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:

    Now evaluate the sub-integral.

  3. 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 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 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:

    The result is:

  4. Now simplify:

  5. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                                                              /                 2   cos(8*x)\
 |                                           3                                                 x*|2*cos(4*x) + 12*x  - --------|
 |    4         4               sin(8*x)    x    sin(4*x)    2 /  sin(4*x)   sin(8*x)   3*x\     \                        8    /
 | sin (x)*x*cos (x)*x dx = C - -------- + --- + -------- + x *|- -------- + -------- + ---| - ---------------------------------
 |                               32768     128     1024        \    128        1024     128/                  512               
/                                                                                                                               
$${{\left(32\,x^2-1\right)\,\sin \left(8\,x\right)+8\,x\,\cos \left(8 \,x\right)+\left(32-256\,x^2\right)\,\sin \left(4\,x\right)-128\,x\, \cos \left(4\,x\right)+256\,x^3}\over{32768}}$$
The answer [src]
      8            8             5       3            7                   2       6            6       2            7                    4       4             3       5   
17*cos (1)   17*sin (1)   329*cos (1)*sin (1)   81*cos (1)*sin(1)   41*cos (1)*sin (1)   41*cos (1)*sin (1)   81*sin (1)*cos(1)   211*cos (1)*sin (1)   329*cos (1)*sin (1)
---------- + ---------- - ------------------- - ----------------- + ------------------ + ------------------ + ----------------- + ------------------- + -------------------
   4096         4096              4096                 4096                1024                 1024                 4096                 2048                  4096       
$${{31\,\sin 8+8\,\cos 8-224\,\sin 4-128\,\cos 4+256}\over{32768}}$$
=
=
      8            8             5       3            7                   2       6            6       2            7                    4       4             3       5   
17*cos (1)   17*sin (1)   329*cos (1)*sin (1)   81*cos (1)*sin(1)   41*cos (1)*sin (1)   41*cos (1)*sin (1)   81*sin (1)*cos(1)   211*cos (1)*sin (1)   329*cos (1)*sin (1)
---------- + ---------- - ------------------- - ----------------- + ------------------ + ------------------ + ----------------- + ------------------- + -------------------
   4096         4096              4096                 4096                1024                 1024                 4096                 2048                  4096       
$$- \frac{329 \sin^{3}{\left(1 \right)} \cos^{5}{\left(1 \right)}}{4096} - \frac{81 \sin{\left(1 \right)} \cos^{7}{\left(1 \right)}}{4096} + \frac{17 \cos^{8}{\left(1 \right)}}{4096} + \frac{41 \sin^{2}{\left(1 \right)} \cos^{6}{\left(1 \right)}}{1024} + \frac{17 \sin^{8}{\left(1 \right)}}{4096} + \frac{81 \sin^{7}{\left(1 \right)} \cos{\left(1 \right)}}{4096} + \frac{41 \sin^{6}{\left(1 \right)} \cos^{2}{\left(1 \right)}}{1024} + \frac{211 \sin^{4}{\left(1 \right)} \cos^{4}{\left(1 \right)}}{2048} + \frac{329 \sin^{5}{\left(1 \right)} \cos^{3}{\left(1 \right)}}{4096}$$
Numerical answer [src]
0.0164397048277092
0.0164397048277092

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