Mister Exam

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Identical expressions
(сos4x-2sin0,5x)
(сos4x minus 2 sinus of 0,5x)
сos4x-2sin0,5x
(сos4x-2sin0,5x)dx
Similar expressions
(сos4x+2sin0,5x)

Integral of (сos4x-2sin0,5x) dx

The solution

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  pi                             
  --                             
  2                              
   /                             
  |                              
  |  (cos(4*x) - 2*sin(0.5*x)) dx
  |                              
 /                               
-pi                              
----                             
 2

$$\int\limits_{- \frac{\pi}{2}}^{\frac{\pi}{2}} \left(- 2 \sin{\left(0.5 x \right)} + \cos{\left(4 x \right)}\right)\, dx$$

Integral(cos(4*x) - 2*sin(0.5*x), (x, -pi/2, pi/2))

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 2 \sin{\left(0.5 x \right)}\right)\, dx = - 2 \int \sin{\left(0.5 x \right)}\, dx$
  1. Let $u = 0.5 x$ .
    
    Then let $du = 0.5 dx$ and substitute $2.0 du$ :
    
    $\int 2.0 \sin{\left(u \right)}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \sin{\left(u \right)}\, du = 2.0 \int \sin{\left(u \right)}\, du$
      1. The integral of sine is negative cosine:
        
        $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
      So, the result is: $- 2.0 \cos{\left(u \right)}$
    Now substitute $u$ back in:
    
    $- 2.0 \cos{\left(0.5 x \right)}$
  So, the result is: $4.0 \cos{\left(0.5 x \right)}$
1. Let $u = 4 x$ .
  
  Then let $du = 4 dx$ and substitute $\frac{du}{4}$ :
  
  $\int \frac{\cos{\left(u \right)}}{4}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{4}$
    1. The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
    So, the result is: $\frac{\sin{\left(u \right)}}{4}$
  Now substitute $u$ back in:
  
  $\frac{\sin{\left(4 x \right)}}{4}$
The result is: $\frac{\sin{\left(4 x \right)}}{4} + 4.0 \cos{\left(0.5 x \right)}$
Add the constant of integration:

$\frac{\sin{\left(4 x \right)}}{4} + 4.0 \cos{\left(0.5 x \right)}+ \mathrm{constant}$

The answer is:

$\frac{\sin{\left(4 x \right)}}{4} + 4.0 \cos{\left(0.5 x \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                            
 |                                    sin(4*x)                 
 | (cos(4*x) - 2*sin(0.5*x)) dx = C + -------- + 4.0*cos(0.5*x)
 |                                       4                     
/

$$\int \left(- 2 \sin{\left(0.5 x \right)} + \cos{\left(4 x \right)}\right)\, dx = C + \frac{\sin{\left(4 x \right)}}{4} + 4.0 \cos{\left(0.5 x \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

$$0$$

Numerical answer [src]

-1.22697953000935e-16

-1.22697953000935e-16

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

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Identical expressions

Similar expressions

Integral of (сos4x-2sin0,5x) dx

Limits of integration:

The graph:

Piecewise:

The solution