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Integral of sin(5x)*sin(7x) 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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 |  sin(5*x)*sin(7*x) dx
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$$\int\limits_{0}^{1} \sin{\left(5 x \right)} \sin{\left(7 x \right)}\, dx$$
Integral(sin(5*x)*sin(7*x), (x, 0, 1))
Detail solution
  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. 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. 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. 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 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:

        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. 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:

            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 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:

      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. 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:

      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. 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:

      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. 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:

      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. 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. 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:

      So, the result is:

    The result is:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                    3     
 |                            sin(4*x)   sin(2*x)   sin (4*x)
 | sin(5*x)*sin(7*x) dx = C - -------- + -------- + ---------
 |                               8          4           6    
/                                                            
$${{\sin \left(2\,x\right)}\over{4}}-{{\sin \left(12\,x\right)}\over{ 24}}$$
The answer [src]
  7*cos(7)*sin(5)   5*cos(5)*sin(7)
- --------------- + ---------------
         24                24      
$$-{{\sin 12-6\,\sin 2}\over{24}}$$
=
=
  7*cos(7)*sin(5)   5*cos(5)*sin(7)
- --------------- + ---------------
         24                24      
$$\frac{5 \sin{\left(7 \right)} \cos{\left(5 \right)}}{24} - \frac{7 \sin{\left(5 \right)} \cos{\left(7 \right)}}{24}$$
Numerical answer [src]
0.249681561623105
0.249681561623105

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