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five - one / two *cos(5x)
5 minus 1 divide by 2 multiply by co sinus of e of (5x)
five minus one divide by two multiply by co sinus of e of (5x)
5-1/2cos(5x)
5-1/2cos5x
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5-1/2*cos(5x)dx
Similar expressions
5+1/2*cos(5x)

Integral of 5-1/2*cos(5x) dx

The solution

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  1                  
  /                  
 |                   
 |  /    cos(5*x)\   
 |  |5 - --------| dx
 |  \       2    /   
 |                   
/                    
0

$$\int\limits_{0}^{1} \left(5 - \frac{\cos{\left(5 x \right)}}{2}\right)\, dx$$

Integral(5 - cos(5*x)/2, (x, 0, 1))

Detail solution

Integrate term-by-term:
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 5\, dx = 5 x$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{\cos{\left(5 x \right)}}{2}\right)\, dx = - \frac{\int \cos{\left(5 x \right)}\, dx}{2}$
  1. Let $u = 5 x$ .
    
    Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
    
    $\int \frac{\cos{\left(u \right)}}{5}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{5}$
      1. The integral of cosine is sine:
        
        $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      So, the result is: $\frac{\sin{\left(u \right)}}{5}$
    Now substitute $u$ back in:
    
    $\frac{\sin{\left(5 x \right)}}{5}$
  So, the result is: $- \frac{\sin{\left(5 x \right)}}{10}$
The result is: $5 x - \frac{\sin{\left(5 x \right)}}{10}$
Add the constant of integration:

$5 x - \frac{\sin{\left(5 x \right)}}{10}+ \mathrm{constant}$

The answer is:

$5 x - \frac{\sin{\left(5 x \right)}}{10}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

    sin(5)
5 - ------
      10

$$5 - \frac{\sin{\left(5 \right)}}{10}$$

    sin(5)
5 - ------
      10

$$5 - \frac{\sin{\left(5 \right)}}{10}$$

5 - sin(5)/10

Numerical answer [src]

5.09589242746631

5.09589242746631

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

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Integral of 5-1/2*cos(5x) dx

Limits of integration:

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Piecewise:

The solution