Integral of (cos2x)/2 dx

The solution

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

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

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:

$\int \frac{\cos{\left(2 x \right)}}{2}\, dx = \frac{\int \cos{\left(2 x \right)}\, dx}{2}$
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:
    
    $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
    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: $\frac{\sin{\left(2 x \right)}}{4}$
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    
 |                           
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

sin(2)
------
  4

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

sin(2)
------
  4

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

sin(2)/4

Numerical answer [src]

0.22732435670642

0.22732435670642

The graph

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

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Integral of (cos2x)/2 dx

Limits of integration:

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Piecewise:

The solution