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Integral of d{x}:
Integral of (4-x^2)^(1/2)
Integral of dx/(1+x^2)
Integral of tan(x)^3
Integral of sin(x)^2*cos(x)^4
Derivative of:
(ctg(x/3))^5
Identical expressions
(ctg(x/ three))^ five
(ctg(x divide by 3)) to the power of 5
(ctg(x divide by three)) to the power of five
(ctg(x/3))5
ctgx/35
(ctg(x/3))⁵
ctgx/3^5
(ctg(x divide by 3))^5
(ctg(x/3))^5dx

Integral of (ctg(x/3))^5 dx

The solution

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  1           
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 |     5/x\   
 |  cot |-| dx
 |      \3/   
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0

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

Integral(cot(x/3)^5, (x, 0, 1))

Detail solution

Rewrite the integrand:

$\cot^{5}{\left(\frac{x}{3} \right)} = \left(\csc^{2}{\left(\frac{x}{3} \right)} - 1\right)^{2} \cot{\left(\frac{x}{3} \right)}$
There are multiple ways to do this integral.
Method #1
1. Let $u = \csc^{2}{\left(\frac{x}{3} \right)}$ .
  
  Then let $du = - \frac{2 \cot{\left(\frac{x}{3} \right)} \csc^{2}{\left(\frac{x}{3} \right)} dx}{3}$ and substitute $- \frac{du}{2}$ :
  
  $\int \left(- \frac{3 u^{2} - 6 u + 3}{2 u}\right)\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{3 u^{2} - 6 u + 3}{u}\, du = - \frac{\int \frac{3 u^{2} - 6 u + 3}{u}\, du}{2}$
    1. Rewrite the integrand:
      
      $\frac{3 u^{2} - 6 u + 3}{u} = 3 u - 6 + \frac{3}{u}$
    2. Integrate term-by-term:
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 3 u\, du = 3 \int u\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
      
      So, the result is: $\frac{3 u^{2}}{2}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int \left(-6\right)\, du = - 6 u$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{3}{u}\, du = 3 \int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      So, the result is: $3 \log{\left(u \right)}$
      
      The result is: $\frac{3 u^{2}}{2} - 6 u + 3 \log{\left(u \right)}$
    So, the result is: $- \frac{3 u^{2}}{4} + 3 u - \frac{3 \log{\left(u \right)}}{2}$
  Now substitute $u$ back in:
  
  $- \frac{3 \log{\left(\csc^{2}{\left(\frac{x}{3} \right)} \right)}}{2} - \frac{3 \csc^{4}{\left(\frac{x}{3} \right)}}{4} + 3 \csc^{2}{\left(\frac{x}{3} \right)}$
Method #2
1. Rewrite the integrand:
  
  $\left(\csc^{2}{\left(\frac{x}{3} \right)} - 1\right)^{2} \cot{\left(\frac{x}{3} \right)} = \cot{\left(\frac{x}{3} \right)} \csc^{4}{\left(\frac{x}{3} \right)} - 2 \cot{\left(\frac{x}{3} \right)} \csc^{2}{\left(\frac{x}{3} \right)} + \cot{\left(\frac{x}{3} \right)}$
2. Integrate term-by-term:
  1. Let $u = \csc{\left(\frac{x}{3} \right)}$ .
    
    Then let $du = - \frac{\cot{\left(\frac{x}{3} \right)} \csc{\left(\frac{x}{3} \right)} dx}{3}$ and substitute $- 3 du$ :
    
    $\int \left(- 3 u^{3}\right)\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int u^{3}\, du = - 3 \int u^{3}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{3}\, du = \frac{u^{4}}{4}$
      
      So, the result is: $- \frac{3 u^{4}}{4}$
    Now substitute $u$ back in:
    
    $- \frac{3 \csc^{4}{\left(\frac{x}{3} \right)}}{4}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 2 \cot{\left(\frac{x}{3} \right)} \csc^{2}{\left(\frac{x}{3} \right)}\right)\, dx = - 2 \int \cot{\left(\frac{x}{3} \right)} \csc^{2}{\left(\frac{x}{3} \right)}\, dx$
    1. Let $u = \csc{\left(\frac{x}{3} \right)}$ .
      
      Then let $du = - \frac{\cot{\left(\frac{x}{3} \right)} \csc{\left(\frac{x}{3} \right)} dx}{3}$ and substitute $- 3 du$ :
      
      $\int \left(- 3 u\right)\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int u\, du = - 3 \int u\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
      
      So, the result is: $- \frac{3 u^{2}}{2}$
      
      Now substitute $u$ back in:
      
      $- \frac{3 \csc^{2}{\left(\frac{x}{3} \right)}}{2}$
    So, the result is: $3 \csc^{2}{\left(\frac{x}{3} \right)}$
  1. Rewrite the integrand:
    
    $\cot{\left(\frac{x}{3} \right)} = \frac{\cos{\left(\frac{x}{3} \right)}}{\sin{\left(\frac{x}{3} \right)}}$
  2. Let $u = \sin{\left(\frac{x}{3} \right)}$ .
    
    Then let $du = \frac{\cos{\left(\frac{x}{3} \right)} dx}{3}$ and substitute $3 du$ :
    
    $\int \frac{3}{u}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{u}\, du = 3 \int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      So, the result is: $3 \log{\left(u \right)}$
    Now substitute $u$ back in:
    
    $3 \log{\left(\sin{\left(\frac{x}{3} \right)} \right)}$
  The result is: $3 \log{\left(\sin{\left(\frac{x}{3} \right)} \right)} - \frac{3 \csc^{4}{\left(\frac{x}{3} \right)}}{4} + 3 \csc^{2}{\left(\frac{x}{3} \right)}$
Method #3
1. Rewrite the integrand:
  
  $\left(\csc^{2}{\left(\frac{x}{3} \right)} - 1\right)^{2} \cot{\left(\frac{x}{3} \right)} = \cot{\left(\frac{x}{3} \right)} \csc^{4}{\left(\frac{x}{3} \right)} - 2 \cot{\left(\frac{x}{3} \right)} \csc^{2}{\left(\frac{x}{3} \right)} + \cot{\left(\frac{x}{3} \right)}$
2. Integrate term-by-term:
  1. Let $u = \csc{\left(\frac{x}{3} \right)}$ .
    
    Then let $du = - \frac{\cot{\left(\frac{x}{3} \right)} \csc{\left(\frac{x}{3} \right)} dx}{3}$ and substitute $- 3 du$ :
    
    $\int \left(- 3 u^{3}\right)\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int u^{3}\, du = - 3 \int u^{3}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{3}\, du = \frac{u^{4}}{4}$
      
      So, the result is: $- \frac{3 u^{4}}{4}$
    Now substitute $u$ back in:
    
    $- \frac{3 \csc^{4}{\left(\frac{x}{3} \right)}}{4}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 2 \cot{\left(\frac{x}{3} \right)} \csc^{2}{\left(\frac{x}{3} \right)}\right)\, dx = - 2 \int \cot{\left(\frac{x}{3} \right)} \csc^{2}{\left(\frac{x}{3} \right)}\, dx$
    1. Let $u = \csc{\left(\frac{x}{3} \right)}$ .
      
      Then let $du = - \frac{\cot{\left(\frac{x}{3} \right)} \csc{\left(\frac{x}{3} \right)} dx}{3}$ and substitute $- 3 du$ :
      
      $\int \left(- 3 u\right)\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int u\, du = - 3 \int u\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
      
      So, the result is: $- \frac{3 u^{2}}{2}$
      
      Now substitute $u$ back in:
      
      $- \frac{3 \csc^{2}{\left(\frac{x}{3} \right)}}{2}$
    So, the result is: $3 \csc^{2}{\left(\frac{x}{3} \right)}$
  1. Rewrite the integrand:
    
    $\cot{\left(\frac{x}{3} \right)} = \frac{\cos{\left(\frac{x}{3} \right)}}{\sin{\left(\frac{x}{3} \right)}}$
  2. Let $u = \sin{\left(\frac{x}{3} \right)}$ .
    
    Then let $du = \frac{\cos{\left(\frac{x}{3} \right)} dx}{3}$ and substitute $3 du$ :
    
    $\int \frac{3}{u}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{u}\, du = 3 \int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      So, the result is: $3 \log{\left(u \right)}$
    Now substitute $u$ back in:
    
    $3 \log{\left(\sin{\left(\frac{x}{3} \right)} \right)}$
  The result is: $3 \log{\left(\sin{\left(\frac{x}{3} \right)} \right)} - \frac{3 \csc^{4}{\left(\frac{x}{3} \right)}}{4} + 3 \csc^{2}{\left(\frac{x}{3} \right)}$
Add the constant of integration:

$- \frac{3 \log{\left(\csc^{2}{\left(\frac{x}{3} \right)} \right)}}{2} - \frac{3 \csc^{4}{\left(\frac{x}{3} \right)}}{4} + 3 \csc^{2}{\left(\frac{x}{3} \right)}+ \mathrm{constant}$

The answer is:

$- \frac{3 \log{\left(\csc^{2}{\left(\frac{x}{3} \right)} \right)}}{2} - \frac{3 \csc^{4}{\left(\frac{x}{3} \right)}}{4} + 3 \csc^{2}{\left(\frac{x}{3} \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                  /   2/x\\        4/x\
 |                              3*log|csc |-||   3*csc |-|
 |    5/x\               2/x\        \    \3//         \3/
 | cot |-| dx = C + 3*csc |-| - -------------- - ---------
 |     \3/                \3/         2              4    
 |                                                        
/

$$\int \cot^{5}{\left(\frac{x}{3} \right)}\, dx = C - \frac{3 \log{\left(\csc^{2}{\left(\frac{x}{3} \right)} \right)}}{2} - \frac{3 \csc^{4}{\left(\frac{x}{3} \right)}}{4} + 3 \csc^{2}{\left(\frac{x}{3} \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

oo

$$\infty$$

oo

$$\infty$$

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Numerical answer [src]

1.76600021982926e+78

1.76600021982926e+78

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

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How to use it?

Identical expressions

Integral of (ctg(x/3))^5 dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3