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Similar expressions
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Solving integrals/ dx/1-cos(2x)

Integral of dx/1-cos(2x) dx

The solution

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

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

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

Detail solution

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

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

$$-{{\sin 2-2}\over{2}}$$

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

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

Simplify

Numerical answer [src]

0.545351286587159

0.545351286587159

The graph

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

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Identical expressions

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Integral of dx/1-cos(2x) dx

Limits of integration:

The graph:

Piecewise:

The solution