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Identical expressions
(four - five *x)cos(x/ two)
(4 minus 5 multiply by x) co sinus of e of (x divide by 2)
(four minus five multiply by x) co sinus of e of (x divide by two)
(4-5x)cos(x/2)
4-5xcosx/2
(4-5*x)cos(x divide by 2)
(4-5*x)cos(x/2)dx
Similar expressions
(4+5*x)cos(x/2)

Integral of (4-5*x)cos(x/2) dx

The solution

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$$\int\limits_{0}^{0} \left(4 - 5 x\right) \cos{\left(\frac{x}{2} \right)}\, dx$$

Integral((4 - 5*x)*cos(x/2), (x, 0, 0))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\left(4 - 5 x\right) \cos{\left(\frac{x}{2} \right)} = - 5 x \cos{\left(\frac{x}{2} \right)} + 4 \cos{\left(\frac{x}{2} \right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 5 x \cos{\left(\frac{x}{2} \right)}\right)\, dx = - 5 \int x \cos{\left(\frac{x}{2} \right)}\, dx$
    1. 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 $\operatorname{dv}{\left(x \right)} = \cos{\left(\frac{x}{2} \right)}$ .
      
      Then $\operatorname{du}{\left(x \right)} = 1$ .
      
      To find $v{\left(x \right)}$ :
      
      Let $u = \frac{x}{2}$ .
      
      Then let $du = \frac{dx}{2}$ and substitute $2 du$ :
      
      $\int 2 \cos{\left(u \right)}\, du$
      
      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:
      
      $2 \sin{\left(\frac{x}{2} \right)}$
      
      Now evaluate the sub-integral.
    2. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 2 \sin{\left(\frac{x}{2} \right)}\, dx = 2 \int \sin{\left(\frac{x}{2} \right)}\, dx$
      
      Let $u = \frac{x}{2}$ .
      
      Then let $du = \frac{dx}{2}$ and substitute $2 du$ :
      
      $\int 2 \sin{\left(u \right)}\, du$
      
      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:
      
      $- 2 \cos{\left(\frac{x}{2} \right)}$
      
      So, the result is: $- 4 \cos{\left(\frac{x}{2} \right)}$
    So, the result is: $- 10 x \sin{\left(\frac{x}{2} \right)} - 20 \cos{\left(\frac{x}{2} \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 4 \cos{\left(\frac{x}{2} \right)}\, dx = 4 \int \cos{\left(\frac{x}{2} \right)}\, dx$
    1. Let $u = \frac{x}{2}$ .
      
      Then let $du = \frac{dx}{2}$ and substitute $2 du$ :
      
      $\int 2 \cos{\left(u \right)}\, du$
      
      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: $8 \sin{\left(\frac{x}{2} \right)}$
  The result is: $- 10 x \sin{\left(\frac{x}{2} \right)} + 8 \sin{\left(\frac{x}{2} \right)} - 20 \cos{\left(\frac{x}{2} \right)}$
Method #2
1. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(x \right)} = 4 - 5 x$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(\frac{x}{2} \right)}$ .
  
  Then $\operatorname{du}{\left(x \right)} = -5$ .
  
  To find $v{\left(x \right)}$ :
  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:
    
    $2 \sin{\left(\frac{x}{2} \right)}$
  Now evaluate the sub-integral.
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 10 \sin{\left(\frac{x}{2} \right)}\right)\, dx = - 10 \int \sin{\left(\frac{x}{2} \right)}\, dx$
  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:
    
    $- 2 \cos{\left(\frac{x}{2} \right)}$
  So, the result is: $20 \cos{\left(\frac{x}{2} \right)}$
Method #3
1. Rewrite the integrand:
  
  $\left(4 - 5 x\right) \cos{\left(\frac{x}{2} \right)} = - 5 x \cos{\left(\frac{x}{2} \right)} + 4 \cos{\left(\frac{x}{2} \right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 5 x \cos{\left(\frac{x}{2} \right)}\right)\, dx = - 5 \int x \cos{\left(\frac{x}{2} \right)}\, dx$
    1. 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 $\operatorname{dv}{\left(x \right)} = \cos{\left(\frac{x}{2} \right)}$ .
      
      Then $\operatorname{du}{\left(x \right)} = 1$ .
      
      To find $v{\left(x \right)}$ :
      
      Let $u = \frac{x}{2}$ .
      
      Then let $du = \frac{dx}{2}$ and substitute $2 du$ :
      
      $\int 2 \cos{\left(u \right)}\, du$
      
      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:
      
      $2 \sin{\left(\frac{x}{2} \right)}$
      
      Now evaluate the sub-integral.
    2. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 2 \sin{\left(\frac{x}{2} \right)}\, dx = 2 \int \sin{\left(\frac{x}{2} \right)}\, dx$
      
      Let $u = \frac{x}{2}$ .
      
      Then let $du = \frac{dx}{2}$ and substitute $2 du$ :
      
      $\int 2 \sin{\left(u \right)}\, du$
      
      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:
      
      $- 2 \cos{\left(\frac{x}{2} \right)}$
      
      So, the result is: $- 4 \cos{\left(\frac{x}{2} \right)}$
    So, the result is: $- 10 x \sin{\left(\frac{x}{2} \right)} - 20 \cos{\left(\frac{x}{2} \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 4 \cos{\left(\frac{x}{2} \right)}\, dx = 4 \int \cos{\left(\frac{x}{2} \right)}\, dx$
    1. Let $u = \frac{x}{2}$ .
      
      Then let $du = \frac{dx}{2}$ and substitute $2 du$ :
      
      $\int 2 \cos{\left(u \right)}\, du$
      
      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: $8 \sin{\left(\frac{x}{2} \right)}$
  The result is: $- 10 x \sin{\left(\frac{x}{2} \right)} + 8 \sin{\left(\frac{x}{2} \right)} - 20 \cos{\left(\frac{x}{2} \right)}$
Add the constant of integration:

$- 10 x \sin{\left(\frac{x}{2} \right)} + 8 \sin{\left(\frac{x}{2} \right)} - 20 \cos{\left(\frac{x}{2} \right)}+ \mathrm{constant}$

The answer is:

$- 10 x \sin{\left(\frac{x}{2} \right)} + 8 \sin{\left(\frac{x}{2} \right)} - 20 \cos{\left(\frac{x}{2} \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                            
 |                                                             
 |              /x\                /x\        /x\           /x\
 | (4 - 5*x)*cos|-| dx = C - 20*cos|-| + 8*sin|-| - 10*x*sin|-|
 |              \2/                \2/        \2/           \2/
 |                                                             
/

$$\int \left(4 - 5 x\right) \cos{\left(\frac{x}{2} \right)}\, dx = C - 10 x \sin{\left(\frac{x}{2} \right)} + 8 \sin{\left(\frac{x}{2} \right)} - 20 \cos{\left(\frac{x}{2} \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

$$0$$

Numerical answer [src]

0.0

0.0

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

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How to use it?

Identical expressions

Similar expressions

Integral of (4-5*x)cos(x/2) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3