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sin^3(x/2)dx

Integral of sin^3(x/2) dx

The solution

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 pi           
  /           
 |            
 |     3/x\   
 |  sin |-| dx
 |      \2/   
 |            
/             
0

$$\int\limits_{0}^{\pi} \sin^{3}{\left(\frac{x}{2} \right)}\, dx$$

Integral(sin(x/2)^3, (x, 0, pi))

Detail solution

Rewrite the integrand:

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

$- \frac{3 \cos{\left(\frac{x}{2} \right)}}{2} + \frac{\cos{\left(\frac{3 x}{2} \right)}}{6}$
Add the constant of integration:

$- \frac{3 \cos{\left(\frac{x}{2} \right)}}{2} + \frac{\cos{\left(\frac{3 x}{2} \right)}}{6}+ \mathrm{constant}$

The answer is:

$- \frac{3 \cos{\left(\frac{x}{2} \right)}}{2} + \frac{\cos{\left(\frac{3 x}{2} \right)}}{6}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                 3/x\
 |                             2*cos |-|
 |    3/x\               /x\         \2/
 | sin |-| dx = C - 2*cos|-| + ---------
 |     \2/               \2/       3    
 |                                      
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

4/3

$$\frac{4}{3}$$

4/3

$$\frac{4}{3}$$

4/3

Numerical answer [src]

1.33333333333333

1.33333333333333

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

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How to use it?

Identical expressions

Integral of sin^3(x/2) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3