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Integral of (x-2)*cos(2*x)*dx 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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 |  (x - 2)*cos(2*x) dx
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$$\int\limits_{0}^{1} \left(x - 2\right) \cos{\left(2 x \right)}\, dx$$
Integral((x - 2)*cos(2*x), (x, 0, 1))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      1. Use integration by parts:

        Let and let .

        Then .

        To find :

        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:

        Now evaluate the sub-integral.

      2. 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. 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:

        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:

    Method #2

    1. Use integration by parts:

      Let and let .

      Then .

      To find :

      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:

      Now evaluate the sub-integral.

    2. 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:

    Method #3

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      1. Use integration by parts:

        Let and let .

        Then .

        To find :

        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:

        Now evaluate the sub-integral.

      2. 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 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:

  2. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                          
 |                                      cos(2*x)   x*sin(2*x)
 | (x - 2)*cos(2*x) dx = C - sin(2*x) + -------- + ----------
 |                                         4           2     
/                                                            
$$\int \left(x - 2\right) \cos{\left(2 x \right)}\, dx = C + \frac{x \sin{\left(2 x \right)}}{2} - \sin{\left(2 x \right)} + \frac{\cos{\left(2 x \right)}}{4}$$
The graph
The answer [src]
  1   sin(2)   cos(2)
- - - ------ + ------
  4     2        4   
$$- \frac{\sin{\left(2 \right)}}{2} - \frac{1}{4} + \frac{\cos{\left(2 \right)}}{4}$$
=
=
  1   sin(2)   cos(2)
- - - ------ + ------
  4     2        4   
$$- \frac{\sin{\left(2 \right)}}{2} - \frac{1}{4} + \frac{\cos{\left(2 \right)}}{4}$$
-1/4 - sin(2)/2 + cos(2)/4
Numerical answer [src]
-0.808685422549626
-0.808685422549626

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