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Identical expressions
(sin^ four (pi*x/ two))
( sinus of to the power of 4( Pi multiply by x divide by 2))
( sinus of to the power of four ( Pi multiply by x divide by two))
(sin4(pi*x/2))
sin4pi*x/2
(sin⁴(pi*x/2))
(sin^4(pix/2))
(sin4(pix/2))
sin4pix/2
sin^4pix/2
(sin^4(pi*x divide by 2))
(sin^4(pi*x/2))dx

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

The solution

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

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

Integral(sin((pi*x)/2)^4, (x, 0, 1))

Detail solution

Rewrite the integrand:

$\sin^{4}{\left(\frac{\pi x}{2} \right)} = \left(\frac{1}{2} - \frac{\cos{\left(\pi x \right)}}{2}\right)^{2}$
There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\left(\frac{1}{2} - \frac{\cos{\left(\pi x \right)}}{2}\right)^{2} = \frac{\cos^{2}{\left(\pi x \right)}}{4} - \frac{\cos{\left(\pi x \right)}}{2} + \frac{1}{4}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\cos^{2}{\left(\pi x \right)}}{4}\, dx = \frac{\int \cos^{2}{\left(\pi x \right)}\, dx}{4}$
    1. Rewrite the integrand:
      
      $\cos^{2}{\left(\pi x \right)} = \frac{\cos{\left(2 \pi x \right)}}{2} + \frac{1}{2}$
    2. Integrate term-by-term:
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\cos{\left(2 \pi x \right)}}{2}\, dx = \frac{\int \cos{\left(2 \pi x \right)}\, dx}{2}$
      
      Let $u = 2 \pi x$ .
      
      Then let $du = 2 \pi dx$ and substitute $\frac{du}{2 \pi}$ :
      
      $\int \frac{\cos{\left(u \right)}}{2 \pi}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{2 \pi}$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $\frac{\sin{\left(u \right)}}{2 \pi}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin{\left(2 \pi x \right)}}{2 \pi}$
      
      So, the result is: $\frac{\sin{\left(2 \pi x \right)}}{4 \pi}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int \frac{1}{2}\, dx = \frac{x}{2}$
      
      The result is: $\frac{x}{2} + \frac{\sin{\left(2 \pi x \right)}}{4 \pi}$
    So, the result is: $\frac{x}{8} + \frac{\sin{\left(2 \pi x \right)}}{16 \pi}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{\cos{\left(\pi x \right)}}{2}\right)\, dx = - \frac{\int \cos{\left(\pi x \right)}\, dx}{2}$
    1. Let $u = \pi x$ .
      
      Then let $du = \pi dx$ and substitute $\frac{du}{\pi}$ :
      
      $\int \frac{\cos{\left(u \right)}}{\pi}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{\pi}$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $\frac{\sin{\left(u \right)}}{\pi}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin{\left(\pi x \right)}}{\pi}$
    So, the result is: $- \frac{\sin{\left(\pi x \right)}}{2 \pi}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int \frac{1}{4}\, dx = \frac{x}{4}$
  The result is: $\frac{3 x}{8} - \frac{\sin{\left(\pi x \right)}}{2 \pi} + \frac{\sin{\left(2 \pi x \right)}}{16 \pi}$
Method #2
1. Rewrite the integrand:
  
  $\left(\frac{1}{2} - \frac{\cos{\left(\pi x \right)}}{2}\right)^{2} = \frac{\cos^{2}{\left(\pi x \right)}}{4} - \frac{\cos{\left(\pi x \right)}}{2} + \frac{1}{4}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\cos^{2}{\left(\pi x \right)}}{4}\, dx = \frac{\int \cos^{2}{\left(\pi x \right)}\, dx}{4}$
    1. Rewrite the integrand:
      
      $\cos^{2}{\left(\pi x \right)} = \frac{\cos{\left(2 \pi x \right)}}{2} + \frac{1}{2}$
    2. Integrate term-by-term:
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\cos{\left(2 \pi x \right)}}{2}\, dx = \frac{\int \cos{\left(2 \pi x \right)}\, dx}{2}$
      
      Let $u = 2 \pi x$ .
      
      Then let $du = 2 \pi dx$ and substitute $\frac{du}{2 \pi}$ :
      
      $\int \frac{\cos{\left(u \right)}}{2 \pi}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{2 \pi}$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $\frac{\sin{\left(u \right)}}{2 \pi}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin{\left(2 \pi x \right)}}{2 \pi}$
      
      So, the result is: $\frac{\sin{\left(2 \pi x \right)}}{4 \pi}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int \frac{1}{2}\, dx = \frac{x}{2}$
      
      The result is: $\frac{x}{2} + \frac{\sin{\left(2 \pi x \right)}}{4 \pi}$
    So, the result is: $\frac{x}{8} + \frac{\sin{\left(2 \pi x \right)}}{16 \pi}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{\cos{\left(\pi x \right)}}{2}\right)\, dx = - \frac{\int \cos{\left(\pi x \right)}\, dx}{2}$
    1. Let $u = \pi x$ .
      
      Then let $du = \pi dx$ and substitute $\frac{du}{\pi}$ :
      
      $\int \frac{\cos{\left(u \right)}}{\pi}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{\pi}$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $\frac{\sin{\left(u \right)}}{\pi}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin{\left(\pi x \right)}}{\pi}$
    So, the result is: $- \frac{\sin{\left(\pi x \right)}}{2 \pi}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int \frac{1}{4}\, dx = \frac{x}{4}$
  The result is: $\frac{3 x}{8} - \frac{\sin{\left(\pi x \right)}}{2 \pi} + \frac{\sin{\left(2 \pi x \right)}}{16 \pi}$
Now simplify:

$\frac{6 \pi x - 8 \sin{\left(\pi x \right)} + \sin{\left(2 \pi x \right)}}{16 \pi}$
Add the constant of integration:

$\frac{6 \pi x - 8 \sin{\left(\pi x \right)} + \sin{\left(2 \pi x \right)}}{16 \pi}+ \mathrm{constant}$

The answer is:

$\frac{6 \pi x - 8 \sin{\left(\pi x \right)} + \sin{\left(2 \pi x \right)}}{16 \pi}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                 
 |                                                  
 |    4/pi*x\          3*x   sin(pi*x)   sin(2*pi*x)
 | sin |----| dx = C + --- - --------- + -----------
 |     \ 2  /           8       2*pi        16*pi   
 |                                                  
/

$$\int \sin^{4}{\left(\frac{\pi x}{2} \right)}\, dx = C + \frac{3 x}{8} - \frac{\sin{\left(\pi x \right)}}{2 \pi} + \frac{\sin{\left(2 \pi x \right)}}{16 \pi}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

3/8

$$\frac{3}{8}$$

3/8

$$\frac{3}{8}$$

3/8

Numerical answer [src]

0.375

0.375

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

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Identical expressions

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

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2