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Integral of d{x}:
Integral of x²dx
Integral of ex
Integral of 1/t
Integral of x/x^2
Identical expressions
sin^ four (x/ two)dx
sinus of to the power of 4(x divide by 2)dx
sinus of to the power of four (x divide by two)dx
sin4(x/2)dx
sin4x/2dx
sin⁴(x/2)dx
sin^4x/2dx
sin^4(x divide by 2)dx

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

The solution

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

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

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

Detail solution

Rewrite the integrand:

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

$\frac{3 x}{8} - \frac{\sin{\left(x \right)}}{2} + \frac{\sin{\left(2 x \right)}}{16}+ \mathrm{constant}$

The answer is:

$\frac{3 x}{8} - \frac{\sin{\left(x \right)}}{2} + \frac{\sin{\left(2 x \right)}}{16}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

3*pi
----
 8

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

3*pi
----
 8

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

3*pi/8

Numerical answer [src]

1.17809724509617

1.17809724509617

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

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Identical expressions

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

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2