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Identical expressions
y=sin⁵(x/ three)cos(x/ three)
y equally sinus of ⁵(x divide by 3) co sinus of e of (x divide by 3)
y equally sinus of ⁵(x divide by three) co sinus of e of (x divide by three)
y=sin⁵x/3cosx/3
y=sin⁵(x divide by 3)cos(x divide by 3)
y=sin⁵(x/3)cos(x/3)dx

Solving integrals/ y=sin⁵(x/3)cos(x/3)

Integral of y=sin⁵(x/3)cos(x/3) dx

The solution

You have entered [src]

  1                  
  /                  
 |                   
 |     5/x\    /x\   
 |  sin |-|*cos|-| dx
 |      \3/    \3/   
 |                   
/                    
0

$$\int\limits_{0}^{1} \sin^{5}{\left(\frac{x}{3} \right)} \cos{\left(\frac{x}{3} \right)}\, dx$$

Simplify

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = \sin{\left(\frac{x}{3} \right)}$ .
  
  Then let $du = \frac{\cos{\left(\frac{x}{3} \right)} dx}{3}$ and substitute $3 du$ :
  
  $\int 9 u^{5}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 3 u^{5}\, du = 3 \int u^{5}\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{5}\, du = \frac{u^{6}}{6}$
    So, the result is: $\frac{u^{6}}{2}$
  Now substitute $u$ back in:
  
  $\frac{\sin^{6}{\left(\frac{x}{3} \right)}}{2}$
Method #2
1. Let $u = \frac{x}{3}$ .
  
  Then let $du = \frac{dx}{3}$ and substitute $3 du$ :
  
  $\int 9 \sin^{5}{\left(u \right)} \cos{\left(u \right)}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 3 \sin^{5}{\left(u \right)} \cos{\left(u \right)}\, du = 3 \int \sin^{5}{\left(u \right)} \cos{\left(u \right)}\, du$
    1. Let $u = \sin{\left(u \right)}$ .
      
      Then let $du = \cos{\left(u \right)} du$ and substitute $du$ :
      
      $\int u^{5}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{5}\, du = \frac{u^{6}}{6}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin^{6}{\left(u \right)}}{6}$
    So, the result is: $\frac{\sin^{6}{\left(u \right)}}{2}$
  Now substitute $u$ back in:
  
  $\frac{\sin^{6}{\left(\frac{x}{3} \right)}}{2}$
Now simplify:

$\frac{\sin^{6}{\left(\frac{x}{3} \right)}}{2}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                           6/x\
 |                         sin |-|
 |    5/x\    /x\              \3/
 | sin |-|*cos|-| dx = C + -------
 |     \3/    \3/             2   
 |                                
/

$${{\sin ^6\left({{x}\over{3}}\right)}\over{2}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

   6     
sin (1/3)
---------
    2

$${{\sin ^6\left({{1}\over{3}}\right)}\over{2}}$$

   6     
sin (1/3)
---------
    2

$$\frac{\sin^{6}{\left(\frac{1}{3} \right)}}{2}$$

Simplify

Numerical answer [src]

0.000613490073607163

0.000613490073607163

The graph

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

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How to use it?

Identical expressions

Integral of y=sin⁵(x/3)cos(x/3) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2