Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of 2/x^4
Integral of (1-x^2)/(1+x^2)
Integral of 1/(1+4x^2)
Integral of x*dx/(x^2+1)
Derivative of:
2ctg(x/2)
Identical expressions
two ctg(x/2)
2ctg(x divide by 2)
two ctg(x divide by 2)
2ctgx/2
2ctg(x/2)dx

Solving integrals/ 2ctg(x/2)

Integral of 2ctg(x/2) dx

The solution

You have entered [src]

  1/5           
   /            
  |             
  |       /x\   
  |  2*cot|-| dx
  |       \2/   
  |             
 /              
3/20

$$\int\limits_{\frac{3}{20}}^{\frac{1}{5}} 2 \cot{\left(\frac{x}{2} \right)}\, dx$$

Integral(2*cot(x/2), (x, 3/20, 1/5))

Detail solution

The integral of a constant times a function is the constant times the integral of the function:

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

$4 \log{\left(\sin{\left(\frac{x}{2} \right)} \right)}$
Add the constant of integration:

$4 \log{\left(\sin{\left(\frac{x}{2} \right)} \right)}+ \mathrm{constant}$

The answer is:

$4 \log{\left(\sin{\left(\frac{x}{2} \right)} \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                               
 |                                
 |      /x\               /   /x\\
 | 2*cot|-| dx = C + 4*log|sin|-||
 |      \2/               \   \2//
 |                                
/

$$4\,\log \sin \left({{x}\over{2}}\right)$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-4*log(sin(3/40)) + 4*log(sin(1/10))

$$4\,\left(\log \sin \left({{1}\over{10}}\right)-\log \sin \left({{3 }\over{40}}\right)\right)$$

-4*log(sin(3/40)) + 4*log(sin(1/10))

$$4 \log{\left(\sin{\left(\frac{1}{10} \right)} \right)} - 4 \log{\left(\sin{\left(\frac{3}{40} \right)} \right)}$$

Упростить

Numerical answer [src]

1.14781010288246

1.14781010288246

The graph

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

Other calculators

How to use it?

Identical expressions

Integral of 2ctg(x/2) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2