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Similar expressions
1/(1-cos(6*x))
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Integral of 1/1-cos(6x) dx

The solution

You have entered [src]

  p                  
  -                  
  6                  
  /                  
 |                   
 |  (1 - cos(6*x)) dx
 |                   
/                    
p                    
-                    
4

$$\int\limits_{\frac{p}{4}}^{\frac{p}{6}} \left(- \cos{\left(6 x \right)} + 1\right)\, dx$$

Integral(1 - cos(6*x), (x, p/4, p/6))

Detail solution

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:
  
  $\int \left(- \cos{\left(6 x \right)}\right)\, dx = - \int \cos{\left(6 x \right)}\, dx$
  1. Let $u = 6 x$ .
    
    Then let $du = 6 dx$ and substitute $\frac{du}{6}$ :
    
    $\int \frac{\cos{\left(u \right)}}{36}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\cos{\left(u \right)}}{6}\, du = \frac{\int \cos{\left(u \right)}\, du}{6}$
      1. The integral of cosine is sine:
        
        $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      So, the result is: $\frac{\sin{\left(u \right)}}{6}$
    Now substitute $u$ back in:
    
    $\frac{\sin{\left(6 x \right)}}{6}$
  So, the result is: $- \frac{\sin{\left(6 x \right)}}{6}$
The result is: $x - \frac{\sin{\left(6 x \right)}}{6}$
Add the constant of integration:

$x - \frac{\sin{\left(6 x \right)}}{6}+ \mathrm{constant}$

The answer is:

$x - \frac{\sin{\left(6 x \right)}}{6}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                    
 |                             sin(6*x)
 | (1 - cos(6*x)) dx = C + x - --------
 |                                6    
/

$$x-{{\sin \left(6\,x\right)}\over{6}}$$

Simplify

The answer [src]

                   /3*p\
                sin|---|
  sin(p)   p       \ 2 /
- ------ - -- + --------
    6      12      6

$${{2\,\sin \left({{3\,p}\over{2}}\right)-2\,\sin p-p}\over{12}}$$

                   /3*p\
                sin|---|
  sin(p)   p       \ 2 /
- ------ - -- + --------
    6      12      6

$$- \frac{p}{12} - \frac{\sin{\left(p \right)}}{6} + \frac{\sin{\left(\frac{3 p}{2} \right)}}{6}$$

Упростить

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

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Integral of 1/1-cos(6x) dx

Limits of integration:

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Piecewise:

The solution