Integral of tg(3*x) dx

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$$\int\limits_{0}^{\frac{\pi}{6}} \tan{\left(3 x \right)}\, dx$$

Integral(tan(3*x), (x, 0, pi/6))

Detail solution

Rewrite the integrand:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

Simplify

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The answer [src]

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Numerical answer [src]

-141.393773941218

-141.393773941218

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Integral of tg(3*x) dx

Limits of integration:

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Piecewise:

The solution

Method #1

Method #2