Mister Exam

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Integral of 1/2cos(x/2)dx dx

The solution

You have entered [src]

 pi          
 --          
 2           
  /          
 |           
 |     /x\   
 |  cos|-|   
 |     \2/   
 |  ------ dx
 |    2      
 |           
/            
0

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

Integral(cos(x/2)/2, (x, 0, pi/2))

Detail solution

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

$\int \frac{\cos{\left(\frac{x}{2} \right)}}{2}\, dx = \frac{\int \cos{\left(\frac{x}{2} \right)}\, dx}{2}$
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: $\sin{\left(\frac{x}{2} \right)}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

  ___
\/ 2 
-----
  2

$$\frac{\sqrt{2}}{2}$$

  ___
\/ 2 
-----
  2

$$\frac{\sqrt{2}}{2}$$

sqrt(2)/2

Numerical answer [src]

0.707106781186547

0.707106781186547

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

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Identical expressions

Integral of 1/2cos(x/2)dx dx

Limits of integration:

The graph:

Piecewise:

The solution