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sin^ three (pi*x/a)
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sin^3(pix/a)
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sin^3(pi*x divide by a)
sin^3(pi*x/a)dx

Integral of sin^3(pi*x/a) dx

The solution

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  a              
  /              
 |               
 |     3/pi*x\   
 |  sin |----| dx
 |      \ a  /   
 |               
/                
0

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

Integral(sin((pi*x)/a)^3, (x, 0, a))

Detail solution

Rewrite the integrand:

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

$\frac{a \left(- 9 \cos{\left(\frac{\pi x}{a} \right)} + \cos{\left(\frac{3 \pi x}{a} \right)}\right)}{12 \pi}$
Add the constant of integration:

$\frac{a \left(- 9 \cos{\left(\frac{\pi x}{a} \right)} + \cos{\left(\frac{3 \pi x}{a} \right)}\right)}{12 \pi}+ \mathrm{constant}$

The answer is:

$\frac{a \left(- 9 \cos{\left(\frac{\pi x}{a} \right)} + \cos{\left(\frac{3 \pi x}{a} \right)}\right)}{12 \pi}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The answer [src]

/4*a                                   
|----  for And(a > -oo, a < oo, a != 0)
<3*pi                                  
|                                      
\ 0               otherwise

$$\begin{cases} \frac{4 a}{3 \pi} & \text{for}\: a > -\infty \wedge a < \infty \wedge a \neq 0 \\0 & \text{otherwise} \end{cases}$$

/4*a                                   
|----  for And(a > -oo, a < oo, a != 0)
<3*pi                                  
|                                      
\ 0               otherwise

$$\begin{cases} \frac{4 a}{3 \pi} & \text{for}\: a > -\infty \wedge a < \infty \wedge a \neq 0 \\0 & \text{otherwise} \end{cases}$$

Piecewise((4*a/(3*pi), (a > -oo)∧(a < oo)∧(Ne(a, 0))), (0, True))

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(pi*x/a) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3