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cos(2x)*cos(7x)

Integral of cos(2x)*cos(7x) dx

Limits of integration:

from to
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The graph:

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Piecewise:

The solution

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 |  cos(2*x)*cos(7*x) dx
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$$\int\limits_{0}^{1} \cos{\left(2 x \right)} \cos{\left(7 x \right)}\, dx$$
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. Let .

            Then let and substitute :

            1. The integral of is when :

            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 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. Let .

              Then let and substitute :

              1. The integral of is when :

              Now substitute back in:

            So, the result is:

          1. The integral of cosine is sine:

          The result is:

        Method #2

        1. Rewrite the integrand:

        2. Integrate term-by-term:

          1. Let .

            Then let and substitute :

            1. The integral of is when :

            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 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. Let .

              Then let and substitute :

              1. The integral of is when :

              Now substitute back in:

            So, the result is:

          1. The integral of cosine is sine:

          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. 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. Let .

            Then let and substitute :

            1. The integral of is when :

            Now substitute back in:

          So, the result is:

        1. The integral of cosine is sine:

        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. Let .

          Then let and substitute :

          1. The integral of is when :

          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 is when :

            Now substitute back in:

          So, the result is:

        1. The integral of cosine is sine:

        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. The integral of cosine is sine:

      So, the result is:

    The result is:

  3. Now simplify:

  4. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                              3             5             9            
 |                                  7      26*sin (x)   128*sin (x)   128*sin (x)         
 | cos(2*x)*cos(7*x) dx = C - 32*sin (x) - ---------- + ----------- + ----------- + sin(x)
 |                                             3             5             9              
/                                                                                         
$${{\sin \left(9\,x\right)}\over{18}}+{{\sin \left(5\,x\right)}\over{ 10}}$$
The graph
The answer [src]
  2*cos(7)*sin(2)   7*cos(2)*sin(7)
- --------------- + ---------------
         45                45      
$${{5\,\sin 9+9\,\sin 5}\over{90}}$$
=
=
  2*cos(7)*sin(2)   7*cos(2)*sin(7)
- --------------- + ---------------
         45                45      
$$\frac{7 \sin{\left(7 \right)} \cos{\left(2 \right)}}{45} - \frac{2 \sin{\left(2 \right)} \cos{\left(7 \right)}}{45}$$
Numerical answer [src]
-0.072996956063994
-0.072996956063994
The graph
Integral of cos(2x)*cos(7x) dx

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