Integral of (1+cos2x)/2 dx

The solution

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

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

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:

$\int \frac{\cos{\left(2 x \right)} + 1}{2}\, dx = \frac{\int \left(\cos{\left(2 x \right)} + 1\right)\, dx}{2}$
1. Integrate term-by-term:
  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:
    
    $\frac{\sin{\left(2 x \right)}}{2}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int 1\, dx = x$
  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    
 |                                   
/

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

Simplify

The graph

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The answer [src]

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

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

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

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

Упростить

Numerical answer [src]

0.72732435670642

0.72732435670642

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

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Integral of (1+cos2x)/2 dx

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The solution