Mister Exam

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Integral of d{x}:
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Integral of 1/(1-exp(y))
Graphing y =:
(1/4)*(1-sin((pi*x)/(2*pi)))
Identical expressions
(one / four)*(one -sin((pi*x)/(two *pi)))
(1 divide by 4) multiply by (1 minus sinus of (( Pi multiply by x) divide by (2 multiply by Pi )))
(one divide by four) multiply by (one minus sinus of (( Pi multiply by x) divide by (two multiply by Pi )))
(1/4)(1-sin((pix)/(2pi)))
1/41-sinpix/2pi
(1 divide by 4)*(1-sin((pi*x) divide by (2*pi)))
(1/4)*(1-sin((pi*x)/(2*pi)))dx
Similar expressions
(1/4)*(1+sin((pi*x)/(2*pi)))

Solving integrals/ (1/4)*(1-sin((pi*x)/(2*pi)))

Integral of (1/4)(1-sin((pix)/(2*pi))) dx

The solution

You have entered [src]

 9*pi                
 ----                
  10                 
   /                 
  |                  
  |         /pi*x\   
  |  1 - sin|----|   
  |         \2*pi/   
  |  ------------- dx
  |        4         
  |                  
 /                   
4*pi                 
----                 
 5

$$\int\limits_{\frac{4 \pi}{5}}^{\frac{9 \pi}{10}} \frac{1 - \sin{\left(\frac{\pi x}{2 \pi} \right)}}{4}\, dx$$

Integral((1 - sin((pi*x)/((2*pi))))/4, (x, 4*pi/5, 9*pi/10))

Detail solution

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

$\int \frac{1 - \sin{\left(\frac{\pi x}{2 \pi} \right)}}{4}\, dx = \frac{\int \left(1 - \sin{\left(\frac{\pi x}{2 \pi} \right)}\right)\, dx}{4}$
1. 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:
    
    $\int \left(- \sin{\left(\frac{\pi x}{2 \pi} \right)}\right)\, dx = - \int \sin{\left(\frac{\pi x}{2 \pi} \right)}\, dx$
    1. Let $u = \frac{\pi x}{2 \pi}$ .
      
      Then let $du = \frac{dx}{2}$ and substitute $2 du$ :
      
      $\int 2 \sin{\left(u \right)}\, du$
      1. The integral of a constant times a function is the constant times the integral of the function:
        
        $\int \sin{\left(u \right)}\, du = 2 \int \sin{\left(u \right)}\, du$
        
        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:
      
      $- 2 \cos{\left(\frac{\pi x}{2 \pi} \right)}$
    So, the result is: $2 \cos{\left(\frac{\pi x}{2 \pi} \right)}$
  The result is: $x + 2 \cos{\left(\frac{\pi x}{2 \pi} \right)}$
So, the result is: $\frac{x}{4} + \frac{\cos{\left(\frac{\pi x}{2 \pi} \right)}}{2}$
Now simplify:

$\frac{x}{4} + \frac{\cos{\left(\frac{x}{2} \right)}}{2}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                                    
 |                                     
 |        /pi*x\             /pi*x\    
 | 1 - sin|----|          cos|----|    
 |        \2*pi/             \2*pi/   x
 | ------------- dx = C + --------- + -
 |       4                    2       4
 |                                     
/

$$\int \frac{1 - \sin{\left(\frac{\pi x}{2 \pi} \right)}}{4}\, dx = C + \frac{x}{4} + \frac{\cos{\left(\frac{\pi x}{2 \pi} \right)}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

                            ___________                    
                           /       ___          /      ___\
                   ___    /  5   \/ 5       ___ |1   \/ 5 |
      ___        \/ 2 *  /   - - -----    \/ 2 *|- + -----|
1   \/ 5    pi         \/    8     8            \4     4  /
- - ----- + -- - ---------------------- + -----------------
8     8     40             4                      4

$$- \frac{\sqrt{5}}{8} - \frac{\sqrt{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{\pi}{40} + \frac{1}{8} + \frac{\sqrt{2} \left(\frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{4}$$

                            ___________                    
                           /       ___          /      ___\
                   ___    /  5   \/ 5       ___ |1   \/ 5 |
      ___        \/ 2 *  /   - - -----    \/ 2 *|- + -----|
1   \/ 5    pi         \/    8     8            \4     4  /
- - ----- + -- - ---------------------- + -----------------
8     8     40             4                      4

$$- \frac{\sqrt{5}}{8} - \frac{\sqrt{2} \sqrt{\frac{5}{8} - \frac{\sqrt{5}}{8}}}{4} + \frac{\pi}{40} + \frac{1}{8} + \frac{\sqrt{2} \left(\frac{1}{4} + \frac{\sqrt{5}}{4}\right)}{4}$$

1/8 - sqrt(5)/8 + pi/40 - sqrt(2)*sqrt(5/8 - sqrt(5)/8)/4 + sqrt(2)*(1/4 + sqrt(5)/4)/4

Numerical answer [src]

0.00224855167238655

0.00224855167238655

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

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Identical expressions

Similar expressions

Integral of (1/4)(1-sin((pix)/(2*pi))) dx

Limits of integration:

The graph:

Piecewise:

The solution

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How to use it?

Identical expressions

Similar expressions

Integral of (1/4)*(1-sin((pi*x)/(2*pi))) dx

Limits of integration:

The graph:

Piecewise:

The solution

Integral of (1/4)(1-sin((pix)/(2*pi))) dx