Integral of tg6x dx

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 |  tan(6*x) dx
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$$\int\limits_{0}^{1} \tan{\left(6 x \right)}\, dx$$

Integral(tan(6*x), (x, 0, 1))

Detail solution

Rewrite the integrand:

$\tan{\left(6 x \right)} = \frac{\sin{\left(6 x \right)}}{\cos{\left(6 x \right)}}$
There are multiple ways to do this integral.
Method #1
1. Let $u = \cos{\left(6 x \right)}$ .
  
  Then let $du = - 6 \sin{\left(6 x \right)} dx$ and substitute $- \frac{du}{6}$ :
  
  $\int \frac{1}{36 u}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{1}{6 u}\right)\, du = - \frac{\int \frac{1}{u}\, du}{6}$
    1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
    So, the result is: $- \frac{\log{\left(u \right)}}{6}$
  Now substitute $u$ back in:
  
  $- \frac{\log{\left(\cos{\left(6 x \right)} \right)}}{6}$
Method #2
1. Let $u = 6 x$ .
  
  Then let $du = 6 dx$ and substitute $\frac{du}{6}$ :
  
  $\int \frac{\sin{\left(u \right)}}{36 \cos{\left(u \right)}}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\sin{\left(u \right)}}{6 \cos{\left(u \right)}}\, du = \frac{\int \frac{\sin{\left(u \right)}}{\cos{\left(u \right)}}\, du}{6}$
    1. Let $u = \cos{\left(u \right)}$ .
      
      Then let $du = - \sin{\left(u \right)} du$ and substitute $- du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{u}\right)\, du = - \int \frac{1}{u}\, du$
      
      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: $- \frac{\log{\left(\cos{\left(u \right)} \right)}}{6}$
  Now substitute $u$ back in:
  
  $- \frac{\log{\left(\cos{\left(6 x \right)} \right)}}{6}$
Add the constant of integration:

$- \frac{\log{\left(\cos{\left(6 x \right)} \right)}}{6}+ \mathrm{constant}$

The answer is:

$- \frac{\log{\left(\cos{\left(6 x \right)} \right)}}{6}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                               
 |                   log(cos(6*x))
 | tan(6*x) dx = C - -------------
 |                         6      
/

$${{\log \sec \left(6\,x\right)}\over{6}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

nan

$$-{{\log \cos 6}\over{6}}$$

=

nan

$$\text{NaN}$$

Simplify

Numerical answer [src]

0.0725238079544744

0.0725238079544744

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Integral of tg6x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2