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Similar expressions
(4x+3)cosx/2

Integral of (4x-3)cosx/2 dx

The solution

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  1                    
  /                    
 |                     
 |  (4*x - 3)*cos(x)   
 |  ---------------- dx
 |         2           
 |                     
/                      
0

$$\int\limits_{0}^{1} \frac{\left(4 x - 3\right) \cos{\left(x \right)}}{2}\, dx$$

Integral(((4*x - 3)*cos(x))/2, (x, 0, 1))

Detail solution

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

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

$2 x \sin{\left(x \right)} - \frac{3 \sin{\left(x \right)}}{2} + 2 \cos{\left(x \right)}+ \mathrm{constant}$

The answer is:

$2 x \sin{\left(x \right)} - \frac{3 \sin{\left(x \right)}}{2} + 2 \cos{\left(x \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

$$\int \frac{\left(4 x - 3\right) \cos{\left(x \right)}}{2}\, dx = C + 2 x \sin{\left(x \right)} - \frac{3 \sin{\left(x \right)}}{2} + 2 \cos{\left(x \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

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

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

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

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

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

Numerical answer [src]

-0.498659895859772

-0.498659895859772

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

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Identical expressions

Similar expressions

Integral of (4x-3)cosx/2 dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2