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(-1)/cos(x)

Integral of (-1)/cos(x) dx

Limits of integration:

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The solution

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  1          
  /          
 |           
 |   -1      
 |  ------ dx
 |  cos(x)   
 |           
/            
0            
$$\int\limits_{0}^{1} \left(- \frac{1}{\cos{\left(x \right)}}\right)\, dx$$
Integral(-1/cos(x), (x, 0, 1))
Detail solution
We have the integral:
  /         
 |          
 |  -1      
 | ------ dx
 | cos(x)   
 |          
/           
The integrand
 -1   
------
cos(x)
Multiply numerator and denominator by
cos(x)
we get
 -1      -cos(x) 
------ = --------
cos(x)      2    
         cos (x) 
Because
sin(a)^2 + cos(a)^2 = 1
then
   2             2   
cos (x) = 1 - sin (x)
transform the denominator
-cos(x)      -cos(x)  
-------- = -----------
   2              2   
cos (x)    1 - sin (x)
do replacement
u = sin(x)
then the integral
  /                
 |                 
 |   -cos(x)       
 | ----------- dx  
 |        2       =
 | 1 - sin (x)     
 |                 
/                  
  
  /                
 |                 
 |   -cos(x)       
 | ----------- dx  
 |        2       =
 | 1 - sin (x)     
 |                 
/                  
  
Because du = dx*cos(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 = sin(x)
The answer
  /                                                   
 |                                                    
 |  -1         log(-1 + sin(x))   log(1 + sin(x))     
 | ------ dx = ---------------- - --------------- + C0
 | cos(x)             2                  2            
 |                                                    
/                                                     
where C0 is constant, independent of x
The answer (Indefinite) [src]
  /                                                  
 |                                                   
 |  -1             log(-1 + sin(x))   log(1 + sin(x))
 | ------ dx = C + ---------------- - ---------------
 | cos(x)                 2                  2       
 |                                                   
/                                                    
$$\int \left(- \frac{1}{\cos{\left(x \right)}}\right)\, dx = C + \frac{\log{\left(\sin{\left(x \right)} - 1 \right)}}{2} - \frac{\log{\left(\sin{\left(x \right)} + 1 \right)}}{2}$$
The graph
The answer [src]
log(1 - sin(1))   log(1 + sin(1))
--------------- - ---------------
       2                 2       
$$\frac{\log{\left(1 - \sin{\left(1 \right)} \right)}}{2} - \frac{\log{\left(\sin{\left(1 \right)} + 1 \right)}}{2}$$
=
=
log(1 - sin(1))   log(1 + sin(1))
--------------- - ---------------
       2                 2       
$$\frac{\log{\left(1 - \sin{\left(1 \right)} \right)}}{2} - \frac{\log{\left(\sin{\left(1 \right)} + 1 \right)}}{2}$$
log(1 - sin(1))/2 - log(1 + sin(1))/2
Numerical answer [src]
-1.22619117088352
-1.22619117088352
The graph
Integral of (-1)/cos(x) dx

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