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Integral of d{x}:
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Derivative of:
sin^3(x/3)
Identical expressions
sin^ three (x/ three)
sinus of cubed (x divide by 3)
sinus of to the power of three (x divide by three)
sin3(x/3)
sin3x/3
sin³(x/3)
sin to the power of 3(x/3)
sin^3x/3
sin^3(x divide by 3)
sin^3(x/3)dx

Integral of sin^3(x/3) dx

The solution

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

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

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

Detail solution

Rewrite the integrand:

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

$- \frac{9 \cos{\left(\frac{x}{3} \right)}}{4} + \frac{\cos{\left(x \right)}}{4}$
Add the constant of integration:

$- \frac{9 \cos{\left(\frac{x}{3} \right)}}{4} + \frac{\cos{\left(x \right)}}{4}+ \mathrm{constant}$

The answer is:

$- \frac{9 \cos{\left(\frac{x}{3} \right)}}{4} + \frac{\cos{\left(x \right)}}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

11/8

$$\frac{11}{8}$$

11/8

$$\frac{11}{8}$$

11/8

Numerical answer [src]

1.375

1.375

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

Other calculators

How to use it?

Identical expressions

Integral of sin^3(x/3) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3