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Integral of d{x}:
Integral of x^3/sqrt(1+x^2)
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Identical expressions
tan^4x/sen^6x
tangent of to the power of 4x divide by sen to the power of 6x
tan4x/sen6x
tan⁴x/sen⁶x
tan^4x divide by sen^6x
tan^4x/sen^6xdx

Solving integrals/ tan^4x/sen^6x

Integral of tan^4x/sen^6x dx

The solution

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  1           
  /           
 |            
 |     4      
 |  tan (x)   
 |  ------- dx
 |     6      
 |  sin (x)   
 |            
/             
0

$$\int\limits_{0}^{1} \frac{\tan^{4}{\left(x \right)}}{\sin^{6}{\left(x \right)}}\, dx$$

Integral(tan(x)^4/(sin(x)^6), (x, 0, 1))

Detail solution

Rewrite the integrand:

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

$\frac{\tan^{3}{\left(x \right)}}{3} + 2 \tan{\left(x \right)} - \cot{\left(x \right)}$
Add the constant of integration:

$\frac{\tan^{3}{\left(x \right)}}{3} + 2 \tan{\left(x \right)} - \cot{\left(x \right)}+ \mathrm{constant}$

The answer is:

$\frac{\tan^{3}{\left(x \right)}}{3} + 2 \tan{\left(x \right)} - \cot{\left(x \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                            
 |                                             
 |    4                                        
 | tan (x)                     2          1    
 | ------- dx = C - cot(x) + ------ + ---------
 |    6                      cot(x)        3   
 | sin (x)                            3*cot (x)
 |                                             
/

$${{\tan ^3x+6\,\tan x}\over{3}}-{{1}\over{\tan x}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

oo

$${\it \%a}$$

oo

$$\infty$$

Упростить

Numerical answer [src]

1.3793236779486e+19

1.3793236779486e+19

The graph

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

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How to use it?

Identical expressions

Integral of tan^4x/sen^6x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3