Mister Exam

Integral of dx/sinx dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1          
  /          
 |           
 |    1      
 |  ------ dx
 |  sin(x)   
 |           
/            
0            
$$\int\limits_{0}^{1} \frac{1}{\sin{\left(x \right)}}\, dx$$
Integral(1/sin(x), (x, 0, 1))
Detail solution
We have the integral:
  /         
 |          
 |   1      
 | ------ dx
 | sin(x)   
 |          
/           
The integrand
  1   
------
sin(x)
Multiply numerator and denominator by
sin(x)
we get
  1       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
 sin(x)      sin(x)  
------- = -----------
   2             2   
sin (x)   1 - cos (x)
do replacement
u = cos(x)
then the integral
  /                
 |                 
 |    sin(x)       
 | ----------- dx  
 |        2       =
 | 1 - cos (x)     
 |                 
/                  
  
  /                
 |                 
 |    sin(x)       
 | ----------- dx  
 |        2       =
 | 1 - cos (x)     
 |                 
/                  
  
Because du = -dx*sin(x)
  /         
 |          
 |  -1      
 | ------ du
 |      2   
 | 1 - u    
 |          
/           
Rewrite the integrand
 -1      -1  /  1       1  \
------ = ---*|----- + -----|
     2    2  \1 - u   1 + u/
1 - u                       
then
                   /             /          
                  |             |           
                  |   1         |   1       
                  | ----- du    | ----- du  
  /               | 1 + u       | 1 - u     
 |                |             |           
 |  -1           /             /           =
 | ------ du = - ----------- - -----------  
 |      2             2             2       
 | 1 - u                                    
 |                                          
/                                           
  
= log(-1 + u)/2 - log(1 + u)/2
do backward replacement
u = cos(x)
The answer
  /                                                   
 |                                                    
 |   1         log(-1 + cos(x))   log(1 + cos(x))     
 | ------ dx = ---------------- - --------------- + C0
 | sin(x)             2                  2            
 |                                                    
/                                                     
where C0 is constant, independent of x
The answer (Indefinite) [src]
  /                                                  
 |                                                   
 |   1             log(-1 + cos(x))   log(1 + cos(x))
 | ------ dx = C + ---------------- - ---------------
 | sin(x)                 2                  2       
 |                                                   
/                                                    
$$\int \frac{1}{\sin{\left(x \right)}}\, dx = C + \frac{\log{\left(\cos{\left(x \right)} - 1 \right)}}{2} - \frac{\log{\left(\cos{\left(x \right)} + 1 \right)}}{2}$$
The graph
The answer [src]
     pi*I
oo + ----
      2  
$$\infty + \frac{i \pi}{2}$$
=
=
     pi*I
oo + ----
      2  
$$\infty + \frac{i \pi}{2}$$
oo + pi*i/2
Numerical answer [src]
44.1790108686112
44.1790108686112
The graph
Integral of dx/sinx dx

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