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Integral of d{x}:
Integral of sec
Integral of x^3*e^x
Integral of (1+x^2)^1/2
Integral of sinx/x
Identical expressions
ctg^ five (x)/sin^ two (x)
ctg to the power of 5(x) divide by sinus of squared (x)
ctg to the power of five (x) divide by sinus of to the power of two (x)
ctg5(x)/sin2(x)
ctg5x/sin2x
ctg⁵(x)/sin²(x)
ctg to the power of 5(x)/sin to the power of 2(x)
ctg^5x/sin^2x
ctg^5(x) divide by sin^2(x)
ctg^5(x)/sin^2(x)dx

Integral of ctg^5(x)/sin^2(x) dx

The solution

You have entered [src]

 pi           
 --           
 3            
  /           
 |            
 |     5      
 |  cot (x)   
 |  ------- dx
 |     2      
 |  sin (x)   
 |            
/             
pi            
--            
4

$$\int\limits_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{\cot^{5}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\, dx$$

Integral(cot(x)^5/sin(x)^2, (x, pi/4, pi/3))

Detail solution

Rewrite the integrand:

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

$- \frac{1}{6 \tan^{6}{\left(x \right)}}$
Add the constant of integration:

$- \frac{1}{6 \tan^{6}{\left(x \right)}}+ \mathrm{constant}$

The answer is:

$- \frac{1}{6 \tan^{6}{\left(x \right)}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                
 |                                3
 |    5             /        2   \ 
 | cot (x)          \-1 + csc (x)/ 
 | ------- dx = C - ---------------
 |    2                    6       
 | sin (x)                         
 |                                 
/

$$\int \frac{\cot^{5}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\, dx = C - \frac{\left(\csc^{2}{\left(x \right)} - 1\right)^{3}}{6}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

13
--
81

$$\frac{13}{81}$$

13
--
81

$$\frac{13}{81}$$

13/81

Numerical answer [src]

0.160493827160494

0.160493827160494

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

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How to use it?

Identical expressions

Integral of ctg^5(x)/sin^2(x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3