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Integral of (1-cos2wt)
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Identical expressions
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Integral of (1-cos2wt) dx

The solution

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  2                    
  /                    
 |                     
 |  (1 - cos(2*w)*t) dw
 |                     
/                      
0

$$\int\limits_{0}^{2} \left(- t \cos{\left(2 w \right)} + 1\right)\, dw$$

Simplify

Detail solution

Integrate term-by-term:
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 1\, dw = w$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- t \cos{\left(2 w \right)}\right)\, dw = - \int t \cos{\left(2 w \right)}\, dw$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int t \cos{\left(2 w \right)}\, dw = t \int \cos{\left(2 w \right)}\, dw$
    1. Let $u = 2 w$ .
      
      Then let $du = 2 dw$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{\cos{\left(u \right)}}{4}\, du$
      1. The integral of a constant times a function is the constant times the integral of the function:
        
        $\int \frac{\cos{\left(u \right)}}{2}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
        
        The integral of cosine is sine:
        
        $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
        
        So, the result is: $\frac{\sin{\left(u \right)}}{2}$
      Now substitute $u$ back in:
      
      $\frac{\sin{\left(2 w \right)}}{2}$
    So, the result is: $\frac{t \sin{\left(2 w \right)}}{2}$
  So, the result is: $- \frac{t \sin{\left(2 w \right)}}{2}$
The result is: $- \frac{t \sin{\left(2 w \right)}}{2} + w$
Add the constant of integration:

$- \frac{t \sin{\left(2 w \right)}}{2} + w+ \mathrm{constant}$

The answer is:

$- \frac{t \sin{\left(2 w \right)}}{2} + w+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                        
 |                               t*sin(2*w)
 | (1 - cos(2*w)*t) dw = C + w - ----------
 |                                   2     
/

$$w-{{t\,\sin \left(2\,w\right)}\over{2}}$$

Simplify

The answer [src]

    t*sin(4)
2 - --------
       2

$$-{{\sin 4\,t-4}\over{2}}$$

    t*sin(4)
2 - --------
       2

$$- \frac{t \sin{\left(4 \right)}}{2} + 2$$

Simplify

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

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Integral of (1-cos2wt) dx

Limits of integration:

The graph:

Piecewise:

The solution