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Integral of 2cos(x/2)dx dx

The solution

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 8*pi           
 ----           
  3             
   /            
  |             
  |       /x\   
  |  2*cos|-| dx
  |       \2/   
  |             
 /              
5*pi            
----            
 3

$$\int\limits_{\frac{5 \pi}{3}}^{\frac{8 \pi}{3}} 2 \cos{\left(\frac{x}{2} \right)}\, dx$$

Integral(2*cos(x/2), (x, 5*pi/3, 8*pi/3))

Detail solution

The integral of a constant times a function is the constant times the integral of the function:

$\int 2 \cos{\left(\frac{x}{2} \right)}\, dx = 2 \int \cos{\left(\frac{x}{2} \right)}\, dx$
1. Let $u = \frac{x}{2}$ .
  
  Then let $du = \frac{dx}{2}$ and substitute $2 du$ :
  
  $\int 2 \cos{\left(u \right)}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \cos{\left(u \right)}\, du = 2 \int \cos{\left(u \right)}\, du$
    1. The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
    So, the result is: $2 \sin{\left(u \right)}$
  Now substitute $u$ back in:
  
  $2 \sin{\left(\frac{x}{2} \right)}$
So, the result is: $4 \sin{\left(\frac{x}{2} \right)}$
Now simplify:

$4 \sin{\left(\frac{x}{2} \right)}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                          
 |                           
 |      /x\               /x\
 | 2*cos|-| dx = C + 4*sin|-|
 |      \2/               \2/
 |                           
/

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

Simplify

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The answer [src]

         ___
-2 - 2*\/ 3

$$- 2 \sqrt{3} - 2$$

         ___
-2 - 2*\/ 3

$$- 2 \sqrt{3} - 2$$

-2 - 2*sqrt(3)

Numerical answer [src]

-5.46410161513776

-5.46410161513776

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

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Integral of 2cos(x/2)dx dx

Limits of integration:

The graph:

Piecewise:

The solution