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3*dx/cos(7*x)

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3*dx/cos(7*x)

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Integral of 3*dx/cos(7*x) dx

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The solution

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  1                
  /                
 |                 
 |         1       
 |  3*1*-------- dx
 |      cos(7*x)   
 |                 
/                  
0                  
$$\int\limits_{0}^{1} 3 \cdot 1 \cdot \frac{1}{\cos{\left(7 x \right)}}\, dx$$
Integral(3*1/cos(7*x), (x, 0, 1))
Detail solution
We have the integral:
  /                 
 |                  
 |          1       
 | 1*3*1*-------- dx
 |       cos(7*x)   
 |                  
/                   
The integrand
       1    
3*1*--------
    cos(7*x)
Multiply numerator and denominator by
cos(7*x)
we get
       1       3*cos(7*x)
3*1*-------- = ----------
    cos(7*x)      2      
               cos (7*x) 
Because
sin(a)^2 + cos(a)^2 = 1
then
   2               2     
cos (7*x) = 1 - sin (7*x)
transform the denominator
3*cos(7*x)     3*cos(7*x) 
---------- = -------------
   2                2     
cos (7*x)    1 - sin (7*x)
do replacement
u = sin(7*x)
then the integral
  /                  
 |                   
 |   3*cos(7*x)      
 | ------------- dx  
 |        2         =
 | 1 - sin (7*x)     
 |                   
/                    
  
  /                  
 |                   
 |   3*cos(7*x)      
 | ------------- dx  
 |        2         =
 | 1 - sin (7*x)     
 |                   
/                    
  
Because du = 7*dx*cos(7*x)
  /             
 |              
 |     3        
 | ---------- du
 |   /     2\   
 | 7*\1 - u /   
 |              
/               
Rewrite the integrand
    3        3*1/7 /  1       1  \
---------- = -----*|----- + -----|
  /     2\     2   \1 - u   1 + u/
7*\1 - u /                        
then
                       /               /          
                      |               |           
                      |   1           |   1       
                   3* | ----- du   3* | ----- du  
  /                   | 1 + u         | 1 - u     
 |                    |               |           
 |     3             /               /           =
 | ---------- du = ------------- + -------------  
 |   /     2\            14              14       
 | 7*\1 - u /                                     
 |                                                
/                                                 
  
= -3*log(-1 + u)/14 + 3*log(1 + u)/14
do backward replacement
u = sin(7*x)
The answer
  /                                                                     
 |                                                                      
 |          1            3*log(-1 + sin(7*x))   3*log(1 + sin(7*x))     
 | 1*3*1*-------- dx = - -------------------- + ------------------- + C0
 |       cos(7*x)                 14                     14             
 |                                                                      
/                                                                       
where C0 is constant, independent of x
The answer (Indefinite) [src]
  /                                                                
 |                                                                 
 |        1              3*log(-1 + sin(7*x))   3*log(1 + sin(7*x))
 | 3*1*-------- dx = C - -------------------- + -------------------
 |     cos(7*x)                   14                     14        
 |                                                                 
/                                                                  
$$\int 3 \cdot 1 \cdot \frac{1}{\cos{\left(7 x \right)}}\, dx = C - \frac{3 \log{\left(\sin{\left(7 x \right)} - 1 \right)}}{14} + \frac{3 \log{\left(\sin{\left(7 x \right)} + 1 \right)}}{14}$$
The graph
The answer [src]
nan
$$\text{NaN}$$
=
=
nan
$$\text{NaN}$$
Numerical answer [src]
1.44728873845077
1.44728873845077
The graph
Integral of 3*dx/cos(7*x) dx

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