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Identical expressions
tg(3lnx- five)/(x)
tg(3lnx minus 5) divide by (x)
tg(3lnx minus five) divide by (x)
tg3lnx-5/x
tg(3lnx-5) divide by (x)
tg(3lnx-5)/(x)dx
Similar expressions
tg(3lnx+5)/(x)

Integral of tg(3lnx-5)/(x) dx

The solution

You have entered [src]

  1                     
  /                     
 |                      
 |  tan(3*log(x) - 5)   
 |  ----------------- dx
 |          x           
 |                      
/                       
0

$$\int\limits_{0}^{1} \frac{\tan{\left(3 \log{\left(x \right)} - 5 \right)}}{x}\, dx$$

Integral(tan(3*log(x) - 5)/x, (x, 0, 1))

Detail solution

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

$- \frac{\log{\left(\cos{\left(3 \log{\left(x \right)} - 5 \right)} \right)}}{3}$
Add the constant of integration:

$- \frac{\log{\left(\cos{\left(3 \log{\left(x \right)} - 5 \right)} \right)}}{3}+ \mathrm{constant}$

The answer is:

$- \frac{\log{\left(\cos{\left(3 \log{\left(x \right)} - 5 \right)} \right)}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                 
 |                                                  
 | tan(3*log(x) - 5)          log(cos(3*log(x) - 5))
 | ----------------- dx = C - ----------------------
 |         x                            3           
 |                                                  
/

$$\int \frac{\tan{\left(3 \log{\left(x \right)} - 5 \right)}}{x}\, dx = C - \frac{\log{\left(\cos{\left(3 \log{\left(x \right)} - 5 \right)} \right)}}{3}$$

Simplify

The answer [src]

  1                      
  /                      
 |                       
 |  tan(-5 + 3*log(x))   
 |  ------------------ dx
 |          x            
 |                       
/                        
0

$$\int\limits_{0}^{1} \frac{\tan{\left(3 \log{\left(x \right)} - 5 \right)}}{x}\, dx$$

  1                      
  /                      
 |                       
 |  tan(-5 + 3*log(x))   
 |  ------------------ dx
 |          x            
 |                       
/                        
0

$$\int\limits_{0}^{1} \frac{\tan{\left(3 \log{\left(x \right)} - 5 \right)}}{x}\, dx$$

Integral(tan(-5 + 3*log(x))/x, (x, 0, 1))

Numerical answer [src]

12.9691735614578

12.9691735614578

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

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Identical expressions

Similar expressions

Integral of tg(3lnx-5)/(x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2