Integral of cot^5xdx dx

The solution

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  1           
  /           
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 |     5      
 |  cot (x) dx
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0

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

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

Detail solution

Rewrite the integrand:

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                                                 
 |                               /   2   \      4   
 |    5                2      log\csc (x)/   csc (x)
 | cot (x) dx = C + csc (x) - ------------ - -------
 |                                 2            4   
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

oo

$$\infty$$

oo

$$\infty$$

oo

Numerical answer [src]

7.26749061658134e+75

7.26749061658134e+75

The graph

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

Other calculators

How to use it?

Identical expressions

Integral of cot^5xdx dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3