Mister Exam

Integral of cos(7x)cos(3x) dx

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The graph:

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Piecewise:

The solution

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 |  cos(7*x)*cos(3*x) dx
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$$\int\limits_{0}^{1} \cos{\left(3 x \right)} \cos{\left(7 x \right)}\, dx$$
Integral(cos(7*x)*cos(3*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. 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:

      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 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:

    The result is:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                                  5     
 |                               3        sin(2*x)   sin(4*x)   4*sin (2*x)
 | cos(7*x)*cos(3*x) dx = C - sin (2*x) + -------- + -------- + -----------
 |                                           4          8            5     
/                                                                          
$${{\sin \left(10\,x\right)}\over{20}}+{{\sin \left(4\,x\right) }\over{8}}$$
The graph
The answer [src]
  3*cos(7)*sin(3)   7*cos(3)*sin(7)
- --------------- + ---------------
         40                40      
$${{2\,\sin 10+5\,\sin 4}\over{40}}$$
=
=
  3*cos(7)*sin(3)   7*cos(3)*sin(7)
- --------------- + ---------------
         40                40      
$$\frac{7 \sin{\left(7 \right)} \cos{\left(3 \right)}}{40} - \frac{3 \sin{\left(3 \right)} \cos{\left(7 \right)}}{40}$$
Numerical answer [src]
-0.12180136745796
-0.12180136745796
The graph
Integral of cos(7x)cos(3x) dx

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