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Solving integrals/ 1/sin4x

Integral of 1/sin4x dx

Limits of integration:

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The solution

You have entered [src] 
  1              
  /              
 |               
 |       1       
 |  1*-------- dx
 |    sin(4*x)   
 |               
/                
0
$$\int\limits_{0}^{1} 1 \cdot \frac{1}{\sin{\left(4 x \right)}}\, dx$$ 
Integral(1/sin(4*x), (x, 0, 1))
Detail solution
We have the integral:
  /               
 |                
 |        1       
 | 1*1*-------- dx
 |     sin(4*x)   
 |                
/
The integrand
     1    
1*--------
  sin(4*x)
Multiply numerator and denominator by
sin(4*x)
we get
     1       1*sin(4*x)
1*-------- = ----------
  sin(4*x)      2      
             sin (4*x)
Because
sin(a)^2 + cos(a)^2 = 1
then
   2               2     
sin (4*x) = 1 - cos (4*x)
transform the denominator
1*sin(4*x)     1*sin(4*x) 
---------- = -------------
   2                2     
sin (4*x)    1 - cos (4*x)
do replacement
u = cos(4*x)
then the integral
  /                  
 |                   
 |   1*sin(4*x)      
 | ------------- dx  
 |        2         =
 | 1 - cos (4*x)     
 |                   
/                    
  
  /                  
 |                   
 |   1*sin(4*x)      
 | ------------- dx  
 |        2         =
 | 1 - cos (4*x)     
 |                   
/
Because du = -4*dx*sin(4*x)
  /             
 |              
 |    -1        
 | ---------- du
 |   /     2\   
 | 4*\1 - u /   
 |              
/
Rewrite the integrand
   -1        1*-1/4 /  1       1  \
---------- = ------*|----- + -----|
  /     2\     2    \1 - u   1 + u/
4*\1 - u /
then
                       /             /          
                      |             |           
                      |   1         |   1       
                      | ----- du    | ----- du  
  /                   | 1 + u       | 1 - u     
 |                    |             |           
 |    -1             /             /           =
 | ---------- du = - ----------- - -----------  
 |   /     2\             8             8       
 | 4*\1 - u /                                   
 |                                              
/                                               
  
= -log(1 + u)/8 + log(-1 + u)/8
do backward replacement
u = cos(4*x)
The answer
  /                                                               
 |                                                                
 |        1            log(1 + cos(4*x))   log(-1 + cos(4*x))     
 | 1*1*-------- dx = - ----------------- + ------------------ + C0
 |     sin(4*x)                8                   8              
 |                                                                
/
where C0 is constant, independent of x
The answer (Indefinite) [src] 
  /                                                          
 |                                                           
 |      1              log(1 + cos(4*x))   log(-1 + cos(4*x))
 | 1*-------- dx = C - ----------------- + ------------------
 |   sin(4*x)                  8                   8         
 |                                                           
/
$${{{{\log \left(\cos \left(4\,x\right)-1\right)}\over{2}}-{{\log \left(\cos \left(4\,x\right)+1\right)}\over{2}}}\over{4}}$$
Simplify
The answer [src] 
nan
$${\it \%a}$$ 
=
= 
nan
$$\text{NaN}$$
Упростить
Numerical answer [src] 
10.3449662462077
10.3449662462077

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

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