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Integral of d{x}:
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Integral of e^x²
Integral of -cos(2x)
Integral of dx/(4x-1)
Identical expressions
(one -cos two x)/2
(1 minus co sinus of e of 2x) divide by 2
(one minus co sinus of e of two x) divide by 2
1-cos2x/2
(1-cos2x) divide by 2
(1-cos2x)/2dx
Similar expressions
(1+cos2x)/2
(1-cos(2*x))/2
(1-cos(2x))/2

Solving integrals/ (1-cos2x)/2

Integral of (1-cos2x)/2 dx

The solution

You have entered [src]

  1                
  /                
 |                 
 |  1 - cos(2*x)   
 |  ------------ dx
 |       2         
 |                 
/                  
0

$$\int\limits_{0}^{1} \frac{1 - \cos{\left(2 x \right)}}{2}\, dx$$

Integral((1 - cos(2*x))/2, (x, 0, 1))

Detail solution

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

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

$\frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4}+ \mathrm{constant}$

The answer is:

$\frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                  
 |                                   
 | 1 - cos(2*x)          x   sin(2*x)
 | ------------ dx = C + - - --------
 |      2                2      4    
 |                                   
/

$$\int \frac{1 - \cos{\left(2 x \right)}}{2}\, dx = C + \frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

1   sin(2)
- - ------
2     4

$$\frac{1}{2} - \frac{\sin{\left(2 \right)}}{4}$$

1   sin(2)
- - ------
2     4

$$\frac{1}{2} - \frac{\sin{\left(2 \right)}}{4}$$

1/2 - sin(2)/4

Numerical answer [src]

0.27267564329358

0.27267564329358

The graph

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

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Integral of (1-cos2x)/2 dx

Limits of integration:

The graph:

Piecewise:

The solution