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Integral of d{x}:
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Integral of dx/x(x-1)
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Identical expressions
(sin(pi*x/ five))^ two
( sinus of ( Pi multiply by x divide by 5)) squared
( sinus of ( Pi multiply by x divide by five)) to the power of two
(sin(pi*x/5))2
sinpi*x/52
(sin(pi*x/5))²
(sin(pi*x/5)) to the power of 2
(sin(pix/5))^2
(sin(pix/5))2
sinpix/52
sinpix/5^2
(sin(pi*x divide by 5))^2
(sin(pi*x/5))^2dx

Solving integrals/ (sin(pi*x/5))^2

Integral of (sin(pi*x/5))^2 dx

The solution

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  5              
  /              
 |               
 |     2/pi*x\   
 |  sin |----| dx
 |      \ 5  /   
 |               
/                
0

$$\int\limits_{0}^{5} \sin^{2}{\left(\frac{\pi x}{5} \right)}\, dx$$

Simplify

Detail solution

Rewrite the integrand:

$\sin^{2}{\left(\frac{\pi x}{5} \right)} = \frac{1}{2} - \frac{\cos{\left(\frac{2 \pi x}{5} \right)}}{2}$
Integrate term-by-term:
1. The integral of a constant is the constant times the variable of integration:
  
  $\int \frac{1}{2}\, dx = \frac{x}{2}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{\cos{\left(\frac{2 \pi x}{5} \right)}}{2}\right)\, dx = - \frac{\int \cos{\left(\frac{2 \pi x}{5} \right)}\, dx}{2}$
  1. Let $u = \frac{2 \pi x}{5}$ .
    
    Then let $du = \frac{2 \pi dx}{5}$ and substitute $\frac{5 du}{2 \pi}$ :
    
    $\int \frac{25 \cos{\left(u \right)}}{4 \pi^{2}}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{5 \cos{\left(u \right)}}{2 \pi}\, du = \frac{5 \int \cos{\left(u \right)}\, du}{2 \pi}$
      1. The integral of cosine is sine:
        
        $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      So, the result is: $\frac{5 \sin{\left(u \right)}}{2 \pi}$
    Now substitute $u$ back in:
    
    $\frac{5 \sin{\left(\frac{2 \pi x}{5} \right)}}{2 \pi}$
  So, the result is: $- \frac{5 \sin{\left(\frac{2 \pi x}{5} \right)}}{4 \pi}$
The result is: $\frac{x}{2} - \frac{5 \sin{\left(\frac{2 \pi x}{5} \right)}}{4 \pi}$
Add the constant of integration:

$\frac{x}{2} - \frac{5 \sin{\left(\frac{2 \pi x}{5} \right)}}{4 \pi}+ \mathrm{constant}$

The answer is:

$\frac{x}{2} - \frac{5 \sin{\left(\frac{2 \pi x}{5} \right)}}{4 \pi}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                             /2*pi*x\
 |                         5*sin|------|
 |    2/pi*x\          x        \  5   /
 | sin |----| dx = C + - - -------------
 |     \ 5  /          2        4*pi    
 |                                      
/

$${{5\,\left({{\pi\,x}\over{5}}-{{\sin \left({{2\,\pi\,x}\over{5}} \right)}\over{2}}\right)}\over{2\,\pi}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

5/2

$$-{{5\,\sin \left(2\,\pi\right)-10\,\pi}\over{4\,\pi}}$$

5/2

$$\frac{5}{2}$$

Simplify

Numerical answer [src]

2.5

2.5

The graph

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

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Identical expressions

Integral of (sin(pi*x/5))^2 dx

Limits of integration:

The graph:

Piecewise:

The solution