Integral of cot³x dx

The solution

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

$$-\log \sin x-{{1}\over{2\,\sin ^2x}}$$

Simplify

The answer [src]

oo

$${\it \%a}$$

oo

$$\infty$$

Упростить

Numerical answer [src]

9.15365037903492e+37

9.15365037903492e+37

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

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Identical expressions

Integral of cot³x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3