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2/sinx
Solving integrals/ 2/sinx

Integral of 2/sinx dx

Limits of integration:

from to
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The graph:

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Piecewise:

The solution

You have entered [src] 
  1          
  /          
 |           
 |    2      
 |  ------ dx
 |  sin(x)   
 |           
/            
0
∫012sin⁡(x) dx\int\limits_{0}^{1} \frac{2}{\sin{\left(x \right)}}\, dx 
Integral(2/sin(x), (x, 0, 1))
Detail solution
We have the integral:
  /         
 |          
 |   2      
 | ------ dx
 | sin(x)   
 |          
/
The integrand
  2   
------
sin(x)
Multiply numerator and denominator by
sin(x)
we get
  2      2*sin(x)
------ = --------
sin(x)      2    
         sin (x)
Because
sin(a)^2 + cos(a)^2 = 1
then
   2             2   
sin (x) = 1 - cos (x)
transform the denominator
2*sin(x)     2*sin(x) 
-------- = -----------
   2              2   
sin (x)    1 - cos (x)
do replacement
u = cos(x)
then the integral
  /                
 |                 
 |   2*sin(x)      
 | ----------- dx  
 |        2       =
 | 1 - cos (x)     
 |                 
/                  
  
  /                
 |                 
 |   2*sin(x)      
 | ----------- dx  
 |        2       =
 | 1 - cos (x)     
 |                 
/
Because du = -dx*sin(x)
  /         
 |          
 |  -2      
 | ------ du
 |      2   
 | 1 - u    
 |          
/
Rewrite the integrand
 -2      2*(-1) /  1       1  \
------ = ------*|----- + -----|
     2     2    \1 - u   1 + u/
1 - u
then
  /                /             /          
 |                |             |           
 |  -2            |   1         |   1       
 | ------ du = -  | ----- du -  | ----- du  
 |      2         | 1 + u       | 1 - u    =
 | 1 - u          |             |           
 |               /             /            
/                                           
  
= -log(1 + u) + log(-1 + u)
do backward replacement
u = cos(x)
The answer
  /                                                    
 |                                                     
 |   2                                                 
 | ------ dx = -log(1 + cos(x)) + log(-1 + cos(x)) + C0
 | sin(x)                                              
 |                                                     
/
where C0 is constant, independent of x
The answer (Indefinite) [src] 
  /                                                  
 |                                                   
 |   2                                               
 | ------ dx = C - log(1 + cos(x)) + log(-1 + cos(x))
 | sin(x)                                            
 |                                                   
/
∫2sin⁡(x) dx=C+log⁡(cos⁡(x)−1)−log⁡(cos⁡(x)+1)\int \frac{2}{\sin{\left(x \right)}}\, dx = C + \log{\left(\cos{\left(x \right)} - 1 \right)} - \log{\left(\cos{\left(x \right)} + 1 \right)}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90025000
Plot the graph f(x)
The answer [src] 
oo + pi*I
∞+iπ\infty + i \pi 
=
= 
oo + pi*I
∞+iπ\infty + i \pi 
oo + pi*i
Numerical answer [src] 
88.3580217372225
88.3580217372225
The graph
Integral of 2/sinx dx

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

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