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Integral of (3-sin(2*x))^2 dx

Limits of integration:

from to
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The graph:

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Piecewise:

The solution

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  1                   
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 |                2   
 |  (3 - sin(2*x))  dx
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$$\int\limits_{2}^{1} \left(3 - \sin{\left(2 x \right)}\right)^{2}\, dx$$
Integral((3 - sin(2*x))^2, (x, 2, 1))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      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:

      1. The integral of a constant times a function is the constant times the integral of the function:

        1. There are multiple ways to do this integral.

          Method #1

          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:

          Method #2

          1. The integral of a constant times a function is the constant times the integral of the function:

            1. There are multiple ways to do this integral.

              Method #1

              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:

              Method #2

              1. Let .

                Then let and substitute :

                1. The integral of is when :

                Now substitute back in:

            So, 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. 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:

      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. The integral of a constant is the constant times the variable of integration:

      The result is:

  2. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                     
 |                                                      
 |               2                       sin(4*x)   19*x
 | (3 - sin(2*x))  dx = C + 3*cos(2*x) - -------- + ----
 |                                          8        2  
/                                                       
$$\int \left(3 - \sin{\left(2 x \right)}\right)^{2}\, dx = C + \frac{19 x}{2} - \frac{\sin{\left(4 x \right)}}{8} + 3 \cos{\left(2 x \right)}$$
The graph
The answer [src]
        2         2                                                                             
     cos (2)   sin (2)      2         2                            cos(2)*sin(2)   cos(4)*sin(4)
-9 + ------- + ------- - cos (4) - sin (4) - 3*cos(4) + 3*cos(2) - ------------- + -------------
        2         2                                                      4               4      
$$-9 + 3 \cos{\left(2 \right)} - \sin^{2}{\left(4 \right)} - \cos^{2}{\left(4 \right)} + \frac{\cos^{2}{\left(2 \right)}}{2} - \frac{\sin{\left(2 \right)} \cos{\left(2 \right)}}{4} + \frac{\sin{\left(4 \right)} \cos{\left(4 \right)}}{4} + \frac{\sin^{2}{\left(2 \right)}}{2} - 3 \cos{\left(4 \right)}$$
=
=
        2         2                                                                             
     cos (2)   sin (2)      2         2                            cos(2)*sin(2)   cos(4)*sin(4)
-9 + ------- + ------- - cos (4) - sin (4) - 3*cos(4) + 3*cos(2) - ------------- + -------------
        2         2                                                      4               4      
$$-9 + 3 \cos{\left(2 \right)} - \sin^{2}{\left(4 \right)} - \cos^{2}{\left(4 \right)} + \frac{\cos^{2}{\left(2 \right)}}{2} - \frac{\sin{\left(2 \right)} \cos{\left(2 \right)}}{4} + \frac{\sin{\left(4 \right)} \cos{\left(4 \right)}}{4} + \frac{\sin^{2}{\left(2 \right)}}{2} - 3 \cos{\left(4 \right)}$$
-9 + cos(2)^2/2 + sin(2)^2/2 - cos(4)^2 - sin(4)^2 - 3*cos(4) + 3*cos(2) - cos(2)*sin(2)/4 + cos(4)*sin(4)/4
Numerical answer [src]
-8.56923955430918
-8.56923955430918

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