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Identical expressions
tg^ three (x/ three)
tg cubed (x divide by 3)
tg to the power of three (x divide by three)
tg3(x/3)
tg3x/3
tg³(x/3)
tg to the power of 3(x/3)
tg^3x/3
tg^3(x divide by 3)
tg^3(x/3)dx

Solving integrals/ tg^3(x/3)

Integral of tg^3(x/3) dx

The solution

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  1           
  /           
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 |     3/x\   
 |  tan |-| dx
 |      \3/   
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/             
0

$$\int\limits_{0}^{1} \tan^{3}{\left(\frac{x}{3} \right)}\, dx$$

Integral(tan(x/3)^3, (x, 0, 1))

Detail solution

Rewrite the integrand:

$\tan^{3}{\left(\frac{x}{3} \right)} = \left(\sec^{2}{\left(\frac{x}{3} \right)} - 1\right) \tan{\left(\frac{x}{3} \right)}$
There are multiple ways to do this integral.
Method #1
1. Let $u = \sec^{2}{\left(\frac{x}{3} \right)}$ .
  
  Then let $du = \frac{2 \tan{\left(\frac{x}{3} \right)} \sec^{2}{\left(\frac{x}{3} \right)} dx}{3}$ and substitute $\frac{du}{2}$ :
  
  $\int \frac{3 u - 3}{4 u}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{3 u - 3}{2 u}\, du = \frac{\int \frac{3 u - 3}{u}\, du}{2}$
    1. Let $u = 3 u$ .
      
      Then let $du = 3 du$ and substitute $du$ :
      
      $\int \frac{u - 3}{u}\, du$
      
      Rewrite the integrand:
      
      $\frac{u - 3}{u} = 1 - \frac{3}{u}$
      
      Integrate term-by-term:
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{3}{u}\right)\, du = - 3 \int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      So, the result is: $- 3 \log{\left(u \right)}$
      
      The result is: $u - 3 \log{\left(u \right)}$
      
      Now substitute $u$ back in:
      
      $3 u - 3 \log{\left(3 u \right)}$
    So, the result is: $\frac{3 u}{2} - \frac{3 \log{\left(3 u \right)}}{2}$
  Now substitute $u$ back in:
  
  $- \frac{3 \log{\left(3 \sec^{2}{\left(\frac{x}{3} \right)} \right)}}{2} + \frac{3 \sec^{2}{\left(\frac{x}{3} \right)}}{2}$
Method #2
1. Rewrite the integrand:
  
  $\left(\sec^{2}{\left(\frac{x}{3} \right)} - 1\right) \tan{\left(\frac{x}{3} \right)} = \tan{\left(\frac{x}{3} \right)} \sec^{2}{\left(\frac{x}{3} \right)} - \tan{\left(\frac{x}{3} \right)}$
2. Integrate term-by-term:
  1. Let $u = \sec^{2}{\left(\frac{x}{3} \right)}$ .
    
    Then let $du = \frac{2 \tan{\left(\frac{x}{3} \right)} \sec^{2}{\left(\frac{x}{3} \right)} dx}{3}$ and substitute $\frac{3 du}{2}$ :
    
    $\int \frac{9}{4}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{3}{2}\, du = \frac{3 \int 1\, du}{2}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      So, the result is: $\frac{3 u}{2}$
    Now substitute $u$ back in:
    
    $\frac{3 \sec^{2}{\left(\frac{x}{3} \right)}}{2}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \tan{\left(\frac{x}{3} \right)}\right)\, dx = - \int \tan{\left(\frac{x}{3} \right)}\, dx$
    1. Rewrite the integrand:
      
      $\tan{\left(\frac{x}{3} \right)} = \frac{\sin{\left(\frac{x}{3} \right)}}{\cos{\left(\frac{x}{3} \right)}}$
    2. Let $u = \cos{\left(\frac{x}{3} \right)}$ .
      
      Then let $du = - \frac{\sin{\left(\frac{x}{3} \right)} dx}{3}$ and substitute $- 3 du$ :
      
      $\int \frac{9}{u}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{3}{u}\right)\, du = - 3 \int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      So, the result is: $- 3 \log{\left(u \right)}$
      
      Now substitute $u$ back in:
      
      $- 3 \log{\left(\cos{\left(\frac{x}{3} \right)} \right)}$
    So, the result is: $3 \log{\left(\cos{\left(\frac{x}{3} \right)} \right)}$
  The result is: $3 \log{\left(\cos{\left(\frac{x}{3} \right)} \right)} + \frac{3 \sec^{2}{\left(\frac{x}{3} \right)}}{2}$
Method #3
1. Rewrite the integrand:
  
  $\left(\sec^{2}{\left(\frac{x}{3} \right)} - 1\right) \tan{\left(\frac{x}{3} \right)} = \tan{\left(\frac{x}{3} \right)} \sec^{2}{\left(\frac{x}{3} \right)} - \tan{\left(\frac{x}{3} \right)}$
2. Integrate term-by-term:
  1. Let $u = \sec^{2}{\left(\frac{x}{3} \right)}$ .
    
    Then let $du = \frac{2 \tan{\left(\frac{x}{3} \right)} \sec^{2}{\left(\frac{x}{3} \right)} dx}{3}$ and substitute $\frac{3 du}{2}$ :
    
    $\int \frac{9}{4}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{3}{2}\, du = \frac{3 \int 1\, du}{2}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      So, the result is: $\frac{3 u}{2}$
    Now substitute $u$ back in:
    
    $\frac{3 \sec^{2}{\left(\frac{x}{3} \right)}}{2}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \tan{\left(\frac{x}{3} \right)}\right)\, dx = - \int \tan{\left(\frac{x}{3} \right)}\, dx$
    1. Rewrite the integrand:
      
      $\tan{\left(\frac{x}{3} \right)} = \frac{\sin{\left(\frac{x}{3} \right)}}{\cos{\left(\frac{x}{3} \right)}}$
    2. Let $u = \cos{\left(\frac{x}{3} \right)}$ .
      
      Then let $du = - \frac{\sin{\left(\frac{x}{3} \right)} dx}{3}$ and substitute $- 3 du$ :
      
      $\int \frac{9}{u}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{3}{u}\right)\, du = - 3 \int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      So, the result is: $- 3 \log{\left(u \right)}$
      
      Now substitute $u$ back in:
      
      $- 3 \log{\left(\cos{\left(\frac{x}{3} \right)} \right)}$
    So, the result is: $3 \log{\left(\cos{\left(\frac{x}{3} \right)} \right)}$
  The result is: $3 \log{\left(\cos{\left(\frac{x}{3} \right)} \right)} + \frac{3 \sec^{2}{\left(\frac{x}{3} \right)}}{2}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                      /     2/x\\        2/x\
 |                  3*log|3*sec |-||   3*sec |-|
 |    3/x\               \      \3//         \3/
 | tan |-| dx = C - ---------------- + ---------
 |     \3/                 2               2    
 |                                              
/

$$3\,\left({{\log \left(\sin ^2\left({{x}\over{3}}\right)-1\right) }\over{2}}-{{1}\over{2\,\sin ^2\left({{x}\over{3}}\right)-2}}\right)$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  3                          3     
- - + 3*log(cos(1/3)) + -----------
  2                          2     
                        2*cos (1/3)

$$3\,\left({{\log \left(1-\sin ^2\left({{1}\over{3}}\right)\right) }\over{2}}-{{1}\over{2\,\sin ^2\left({{1}\over{3}}\right)-2}}-{{1 }\over{2}}\right)$$

  3                          3     
- - + 3*log(cos(1/3)) + -----------
  2                          2     
                        2*cos (1/3)

$$- \frac{3}{2} + 3 \log{\left(\cos{\left(\frac{1}{3} \right)} \right)} + \frac{3}{2 \cos^{2}{\left(\frac{1}{3} \right)}}$$

Упростить

Numerical answer [src]

0.00998954487832324

0.00998954487832324

The graph

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

Other calculators

How to use it?

Identical expressions

Integral of tg^3(x/3) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3