Integral of ctg(2x-1) dx

The solution

You have entered [src]

  7                
  /                
 |                 
 |  cot(2*x - 1) dx
 |                 
/                  
5

$$\int\limits_{5}^{7} \cot{\left(2 x - 1 \right)}\, dx$$

Simplify

Detail solution

Rewrite the integrand:

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

$\frac{\log{\left(\sin{\left(2 x - 1 \right)} \right)}}{2}$
Add the constant of integration:

$\frac{\log{\left(\sin{\left(2 x - 1 \right)} \right)}}{2}+ \mathrm{constant}$

The answer is:

$\frac{\log{\left(\sin{\left(2 x - 1 \right)} \right)}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                       
 |                       log(sin(2*x - 1))
 | cot(2*x - 1) dx = C + -----------------
 |                               2        
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

                                 /       2    \      /       2   \       
log(tan(13))   log(-tan(9))   log\1 + tan (13)/   log\1 + tan (9)/   pi*I
------------ - ------------ - ----------------- + ---------------- - ----
     2              2                 4                  4            2

$${{\log \sin 13-\log \sin 9}\over{2}}$$

                                 /       2    \      /       2   \       
log(tan(13))   log(-tan(9))   log\1 + tan (13)/   log\1 + tan (9)/   pi*I
------------ - ------------ - ----------------- + ---------------- - ----
     2              2                 4                  4            2

$$\frac{\log{\left(\tan{\left(13 \right)} \right)}}{2} - \frac{\log{\left(\tan^{2}{\left(13 \right)} + 1 \right)}}{4} + \frac{\log{\left(\tan^{2}{\left(9 \right)} + 1 \right)}}{4} - \frac{\log{\left(- \tan{\left(9 \right)} \right)}}{2} - \frac{i \pi}{2}$$

Simplify

Numerical answer [src]

-2.42618972318683

-2.42618972318683

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 ctg(2x-1) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2