Integral of -cos2x/2 dx

The solution

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

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

Integral(-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:

$\int \frac{\left(-1\right) \cos{\left(2 x \right)}}{2}\, dx = \frac{\int \left(- \cos{\left(2 x \right)}\right)\, dx}{2}$
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: $- \frac{\sin{\left(2 x \right)}}{2}$
So, the result is: $- \frac{\sin{\left(2 x \right)}}{4}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

$$-{{\sin \left(2\,x\right)}\over{4}}$$

Simplify

The graph

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The answer [src]

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

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

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

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

Упростить

Numerical answer [src]

-0.22732435670642

-0.22732435670642

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

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Integral of -cos2x/2 dx

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The solution