Integral of tg(2x) dx

The solution

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 |  tan(2*x) dx
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$$\int\limits_{0}^{1} \tan{\left(2 x \right)}\, dx$$

Integral(tan(2*x), (x, 0, 1))

Detail solution

Rewrite the integrand:

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                               
 |                   log(cos(2*x))
 | tan(2*x) dx = C - -------------
 |                         2      
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$$\int \tan{\left(2 x \right)}\, dx = C - \frac{\log{\left(\cos{\left(2 x \right)} \right)}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

nan

$$\text{NaN}$$

nan

$$\text{NaN}$$

nan

Numerical answer [src]

-0.96117546625349

-0.96117546625349

The graph

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of tg(2x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2