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Solving integrals/ (3x-2)*cos(x/2)

Integral of (3x-2)*cos(x/2) dx

The solution

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0

$$\int\limits_{0}^{1} \left(3 x - 2\right) \cos{\left(\frac{x}{2} \right)}\, dx$$

Integral((3*x - 1*2)*cos(x/2), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\left(3 x - 2\right) \cos{\left(\frac{x}{2} \right)} = 3 x \cos{\left(\frac{x}{2} \right)} - 2 \cos{\left(\frac{x}{2} \right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 3 x \cos{\left(\frac{x}{2} \right)}\, dx = 3 \int x \cos{\left(\frac{x}{2} \right)}\, dx$
    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 $2 du$ :
      
      $\int 4 \cos{\left(u \right)}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 2 \cos{\left(u \right)}\, du = 2 \int \cos{\left(u \right)}\, du$
      
      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 $2 du$ :
      
      $\int 4 \sin{\left(u \right)}\, du$
      
      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: $- 2 \cos{\left(u \right)}$
      
      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: $6 x \sin{\left(\frac{x}{2} \right)} + 12 \cos{\left(\frac{x}{2} \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 2 \cos{\left(\frac{x}{2} \right)}\right)\, dx = - 2 \int \cos{\left(\frac{x}{2} \right)}\, dx$
    1. Let $u = \frac{x}{2}$ .
      
      Then let $du = \frac{dx}{2}$ and substitute $2 du$ :
      
      $\int 4 \cos{\left(u \right)}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 2 \cos{\left(u \right)}\, du = 2 \int \cos{\left(u \right)}\, du$
      
      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: $- 4 \sin{\left(\frac{x}{2} \right)}$
  The result is: $6 x \sin{\left(\frac{x}{2} \right)} - 4 \sin{\left(\frac{x}{2} \right)} + 12 \cos{\left(\frac{x}{2} \right)}$
Method #2
1. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(x \right)} = 3 x - 2$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(\frac{x}{2} \right)}$ .
  
  Then $\operatorname{du}{\left(x \right)} = 3$ .
  
  To find $v{\left(x \right)}$ :
  1. Let $u = \frac{x}{2}$ .
    
    Then let $du = \frac{dx}{2}$ and substitute $2 du$ :
    
    $\int 4 \cos{\left(u \right)}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 2 \cos{\left(u \right)}\, du = 2 \int \cos{\left(u \right)}\, du$
      
      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:
  
  $\int 6 \sin{\left(\frac{x}{2} \right)}\, dx = 6 \int \sin{\left(\frac{x}{2} \right)}\, dx$
  1. Let $u = \frac{x}{2}$ .
    
    Then let $du = \frac{dx}{2}$ and substitute $2 du$ :
    
    $\int 4 \sin{\left(u \right)}\, du$
    1. 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: $- 2 \cos{\left(u \right)}$
    Now substitute $u$ back in:
    
    $- 2 \cos{\left(\frac{x}{2} \right)}$
  So, the result is: $- 12 \cos{\left(\frac{x}{2} \right)}$
Method #3
1. Rewrite the integrand:
  
  $\left(3 x - 2\right) \cos{\left(\frac{x}{2} \right)} = 3 x \cos{\left(\frac{x}{2} \right)} - 2 \cos{\left(\frac{x}{2} \right)}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 3 x \cos{\left(\frac{x}{2} \right)}\, dx = 3 \int x \cos{\left(\frac{x}{2} \right)}\, dx$
    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 $2 du$ :
      
      $\int 4 \cos{\left(u \right)}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 2 \cos{\left(u \right)}\, du = 2 \int \cos{\left(u \right)}\, du$
      
      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 $2 du$ :
      
      $\int 4 \sin{\left(u \right)}\, du$
      
      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: $- 2 \cos{\left(u \right)}$
      
      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: $6 x \sin{\left(\frac{x}{2} \right)} + 12 \cos{\left(\frac{x}{2} \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 2 \cos{\left(\frac{x}{2} \right)}\right)\, dx = - 2 \int \cos{\left(\frac{x}{2} \right)}\, dx$
    1. Let $u = \frac{x}{2}$ .
      
      Then let $du = \frac{dx}{2}$ and substitute $2 du$ :
      
      $\int 4 \cos{\left(u \right)}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 2 \cos{\left(u \right)}\, du = 2 \int \cos{\left(u \right)}\, du$
      
      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: $- 4 \sin{\left(\frac{x}{2} \right)}$
  The result is: $6 x \sin{\left(\frac{x}{2} \right)} - 4 \sin{\left(\frac{x}{2} \right)} + 12 \cos{\left(\frac{x}{2} \right)}$
Add the constant of integration:

$6 x \sin{\left(\frac{x}{2} \right)} - 4 \sin{\left(\frac{x}{2} \right)} + 12 \cos{\left(\frac{x}{2} \right)}+ \mathrm{constant}$

The answer is:

$6 x \sin{\left(\frac{x}{2} \right)} - 4 \sin{\left(\frac{x}{2} \right)} + 12 \cos{\left(\frac{x}{2} \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                           
 |                                                            
 |              /x\               /x\         /x\          /x\
 | (3*x - 2)*cos|-| dx = C - 4*sin|-| + 12*cos|-| + 6*x*sin|-|
 |              \2/               \2/         \2/          \2/
 |                                                            
/

$$2\,\left(6\,\left({{\sin \left({{x}\over{2}}\right)\,x}\over{2}}+ \cos \left({{x}\over{2}}\right)\right)-2\,\sin \left({{x}\over{2}} \right)\right)$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-12 + 2*sin(1/2) + 12*cos(1/2)

$$2\,\sin \left({{1}\over{2}}\right)+12\,\cos \left({{1}\over{2}} \right)-12$$

-12 + 2*sin(1/2) + 12*cos(1/2)

$$-12 + 2 \sin{\left(\frac{1}{2} \right)} + 12 \cos{\left(\frac{1}{2} \right)}$$

Упростить

Numerical answer [src]

-0.510158180107121

-0.510158180107121

The graph

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

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Identical expressions

Similar expressions

Integral of (3x-2)*cos(x/2) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3