Integral of tan(x/2) dx

The solution

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

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

Detail solution

Rewrite the integrand:

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

$- 2 \log{\left(\cos{\left(\frac{x}{2} \right)} \right)}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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$$2\,\log \sec \left({{x}\over{2}}\right)$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

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

$$-2\,\log \cos \left({{1}\over{2}}\right)$$

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

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

Упростить

Numerical answer [src]

0.261168480887445

0.261168480887445

The graph

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

Other calculators

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Identical expressions

Similar expressions

Integral of tan(x/2) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2