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Solving integrals/ (2-x)sinx/2

Integral of (2-x)sinx/2 dx

The solution

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

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

Integral(2 - x*sin(x)/2, (x, 0, 1))

Detail solution

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

$\int \frac{\left(2 - x\right) \sin{\left(x \right)}}{2}\, dx = \frac{\int \left(2 - x\right) \sin{\left(x \right)}\, dx}{2}$
1. There are multiple ways to do this integral.
  Method #1
  1. Let $u = - x$ .
    
    Then let $du = - dx$ and substitute $du$ :
    
    $\int \left(u \sin{\left(u \right)} + 2 \sin{\left(u \right)}\right)\, du$
    1. Integrate term-by-term:
      
      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 $v{\left(u \right)}$ :
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
      
      Now evaluate the sub-integral.
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \cos{\left(u \right)}\right)\, du = - \int \cos{\left(u \right)}\, du$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $- \sin{\left(u \right)}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 2 \sin{\left(u \right)}\, du = 2 \int \sin{\left(u \right)}\, du$
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
      
      So, the result is: $- 2 \cos{\left(u \right)}$
      
      The result is: $- u \cos{\left(u \right)} + \sin{\left(u \right)} - 2 \cos{\left(u \right)}$
    Now substitute $u$ back in:
    
    $x \cos{\left(x \right)} - \sin{\left(x \right)} - 2 \cos{\left(x \right)}$
  Method #2
  1. Rewrite the integrand:
    
    $\left(2 - x\right) \sin{\left(x \right)} = - x \sin{\left(x \right)} + 2 \sin{\left(x \right)}$
  2. Integrate term-by-term:
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- x \sin{\left(x \right)}\right)\, dx = - \int x \sin{\left(x \right)}\, dx$
      
      Use integration by parts:
      
      $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
      
      Let $u{\left(x \right)} = x$ and let $\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)}$ .
      
      Then $\operatorname{du}{\left(x \right)} = 1$ .
      
      To find $v{\left(x \right)}$ :
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
      
      Now evaluate the sub-integral.
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \cos{\left(x \right)}\right)\, dx = - \int \cos{\left(x \right)}\, dx$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
      
      So, the result is: $- \sin{\left(x \right)}$
      
      So, the result is: $x \cos{\left(x \right)} - \sin{\left(x \right)}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 2 \sin{\left(x \right)}\, dx = 2 \int \sin{\left(x \right)}\, dx$
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
      
      So, the result is: $- 2 \cos{\left(x \right)}$
    The result is: $x \cos{\left(x \right)} - \sin{\left(x \right)} - 2 \cos{\left(x \right)}$
  Method #3
  1. Use integration by parts:
    
    $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
    
    Let $u{\left(x \right)} = 2 - x$ and let $\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)}$ .
    
    Then $\operatorname{du}{\left(x \right)} = -1$ .
    
    To find $v{\left(x \right)}$ :
    1. The integral of sine is negative cosine:
      
      $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
    Now evaluate the sub-integral.
  2. The integral of cosine is sine:
    
    $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
So, the result is: $\frac{x \cos{\left(x \right)}}{2} - \frac{\sin{\left(x \right)}}{2} - \cos{\left(x \right)}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                                                  
 |                                                   
 | (2 - x)*sin(x)                   sin(x)   x*cos(x)
 | -------------- dx = C - cos(x) - ------ + --------
 |       2                            2         2    
 |                                                   
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

    cos(1)   sin(1)
1 - ------ - ------
      2        2

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

    cos(1)   sin(1)
1 - ------ - ------
      2        2

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

Упростить

Numerical answer [src]

0.309113354661982

0.309113354661982

The graph

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

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Identical expressions

Similar expressions

Integral of (2-x)sinx/2 dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3