Mister Exam

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Integral of d{x}:
Integral of 1/√(1-4x^2)
Integral of x/(x^2+2x+2)
Integral of x^2e^x
Integral of x^2/(x-2)
Identical expressions
cosec^ four (x)
co sinus of e of ec to the power of 4(x)
co sinus of e of ec to the power of four (x)
cosec4(x)
cosec4x
cosec⁴(x)
cosec^4x
cosec^4(x)dx

Integral of cosec^4(x) dx

The solution

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  1           
  /           
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 |     4      
 |  csc (x) dx
 |            
/             
0

$$\int\limits_{0}^{1} \csc^{4}{\left(x \right)}\, dx$$

Integral(csc(x)^4, (x, 0, 1))

Detail solution

Rewrite the integrand:

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

$- \frac{\cot^{3}{\left(x \right)}}{3} - \cot{\left(x \right)}+ \mathrm{constant}$

The answer is:

$- \frac{\cot^{3}{\left(x \right)}}{3} - \cot{\left(x \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                 
 |                              3   
 |    4                      cot (x)
 | csc (x) dx = C - cot(x) - -------
 |                              3   
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

oo

$$\infty$$

oo

$$\infty$$

oo

Numerical answer [src]

7.81431122445857e+56

7.81431122445857e+56

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

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How to use it?

Identical expressions

Integral of cosec^4(x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3