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Solving integrals/ 1-cos((pi*x)/2)

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

The solution

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  4                   
  /                   
 |                    
 |  /       /pi*x\\   
 |  |1 - cos|----|| dx
 |  \       \ 2  //   
 |                    
/                     
0

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

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

Detail solution

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                                  /pi*x\
 |                              2*sin|----|
 | /       /pi*x\\                   \ 2  /
 | |1 - cos|----|| dx = C + x - -----------
 | \       \ 2  //                   pi    
 |                                         
/

$$\int \left(1 - \cos{\left(\frac{\pi x}{2} \right)}\right)\, dx = C + x - \frac{2 \sin{\left(\frac{\pi x}{2} \right)}}{\pi}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

$$4$$

Упростить

Numerical answer [src]

4.0

4.0

The graph

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

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

Limits of integration:

The graph:

Piecewise:

The solution