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Identical expressions
cot^ three (x)
cotangent of cubed (x)
cotangent of to the power of three (x)
cot3(x)
cot3x
cot³(x)
cot to the power of 3(x)
cot^3x
cot^3(x)dx

Integral of cot^3(x) dx

The solution

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  1           
  /           
 |            
 |     3      
 |  cot (x) dx
 |            
/             
0

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

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

Detail solution

Rewrite the integrand:

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                                       
 |                     /   2   \      2   
 |    3             log\csc (x)/   csc (x)
 | cot (x) dx = C + ------------ - -------
 |                       2            2   
/

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

Simplify

The answer [src]

oo

$$\infty$$

oo

$$\infty$$

oo

Numerical answer [src]

9.15365037903492e+37

9.15365037903492e+37

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

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How to use it?

Identical expressions

Integral of cot^3(x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3