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1/4xsin(x/2)dx

Integral of 1/4xsin(x/2) dx

The solution

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 2*pi           
   /            
  |             
  |  x    /x\   
  |  -*sin|-| dx
  |  4    \2/   
  |             
 /              
 0

$$\int\limits_{0}^{2 \pi} \frac{x}{4} \sin{\left(\frac{x}{2} \right)}\, dx$$

Integral((x/4)*sin(x/2), (x, 0, 2*pi))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = \frac{x}{2}$ .
  
  Then let $du = \frac{dx}{2}$ and substitute $du$ :
  
  $\int u \sin{\left(u \right)}\, du$
  1. Use integration by parts:
    
    $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
    
    Let $u{\left(u \right)} = u$ and let $\operatorname{dv}{\left(u \right)} = \sin{\left(u \right)}$ .
    
    Then $\operatorname{du}{\left(u \right)} = 1$ .
    
    To find $v{\left(u \right)}$ :
    1. The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
    Now evaluate the sub-integral.
  2. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \cos{\left(u \right)}\right)\, du = - \int \cos{\left(u \right)}\, du$
    1. The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
    So, the result is: $- \sin{\left(u \right)}$
  Now substitute $u$ back in:
  
  $- \frac{x \cos{\left(\frac{x}{2} \right)}}{2} + \sin{\left(\frac{x}{2} \right)}$
Method #2
1. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(x \right)} = \frac{x}{4}$ and let $\operatorname{dv}{\left(x \right)} = \sin{\left(\frac{x}{2} \right)}$ .
  
  Then $\operatorname{du}{\left(x \right)} = \frac{1}{4}$ .
  
  To find $v{\left(x \right)}$ :
  1. Let $u = \frac{x}{2}$ .
    
    Then let $du = \frac{dx}{2}$ and substitute $2 du$ :
    
    $\int 2 \sin{\left(u \right)}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \sin{\left(u \right)}\, du = 2 \int \sin{\left(u \right)}\, du$
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
      
      So, the result is: $- 2 \cos{\left(u \right)}$
    Now substitute $u$ back in:
    
    $- 2 \cos{\left(\frac{x}{2} \right)}$
  Now evaluate the sub-integral.
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{\cos{\left(\frac{x}{2} \right)}}{2}\right)\, dx = - \frac{\int \cos{\left(\frac{x}{2} \right)}\, dx}{2}$
  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:
      
      $\int \cos{\left(u \right)}\, du = 2 \int \cos{\left(u \right)}\, du$
      
      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:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

pi

$$\pi$$

pi

$$\pi$$

pi

Numerical answer [src]

3.14159265358979

3.14159265358979

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

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How to use it?

Identical expressions

Integral of 1/4xsin(x/2) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2