Mister Exam

Other calculators

1/cos3x
Solving integrals/ 1/cos3x

Integral of 1/cos3x dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1            
  /            
 |             
 |     1       
 |  -------- dx
 |  cos(3*x)   
 |             
/              
0
$$\int\limits_{0}^{1} \frac{1}{\cos{\left(3 x \right)}}\, dx$$ 
Integral(1/cos(3*x), (x, 0, 1))
Detail solution
We have the integral:
  /           
 |            
 |    1       
 | -------- dx
 | cos(3*x)   
 |            
/
The integrand
   1    
--------
cos(3*x)
Multiply numerator and denominator by
cos(3*x)
we get
   1        cos(3*x)
-------- = ---------
cos(3*x)      2     
           cos (3*x)
Because
sin(a)^2 + cos(a)^2 = 1
then
   2               2     
cos (3*x) = 1 - sin (3*x)
transform the denominator
 cos(3*x)      cos(3*x)  
--------- = -------------
   2               2     
cos (3*x)   1 - sin (3*x)
do replacement
u = sin(3*x)
then the integral
  /                  
 |                   
 |    cos(3*x)       
 | ------------- dx  
 |        2         =
 | 1 - sin (3*x)     
 |                   
/                    
  
  /                  
 |                   
 |    cos(3*x)       
 | ------------- dx  
 |        2         =
 | 1 - sin (3*x)     
 |                   
/
Because du = 3*dx*cos(3*x)
  /             
 |              
 |     1        
 | ---------- du
 |   /     2\   
 | 3*\1 - u /   
 |              
/
Rewrite the integrand
    1         1  /  1       1  \
---------- = ---*|----- + -----|
  /     2\   3*2 \1 - u   1 + u/
3*\1 - u /
then
                     /             /          
                    |             |           
                    |   1         |   1       
                    | ----- du    | ----- du  
  /                 | 1 + u       | 1 - u     
 |                  |             |           
 |     1           /             /           =
 | ---------- du = ----------- + -----------  
 |   /     2\           6             6       
 | 3*\1 - u /                                 
 |                                            
/                                             
  
= -log(-1 + u)/6 + log(1 + u)/6
do backward replacement
u = sin(3*x)
The answer
  /                                                           
 |                                                            
 |    1            log(-1 + sin(3*x))   log(1 + sin(3*x))     
 | -------- dx = - ------------------ + ----------------- + C0
 | cos(3*x)                6                    6             
 |                                                            
/
where C0 is constant, independent of x
The answer (Indefinite) [src] 
  /                                                        
 |                                                         
 |    1              log(-1 + sin(3*x))   log(1 + sin(3*x))
 | -------- dx = C - ------------------ + -----------------
 | cos(3*x)                  6                    6        
 |                                                         
/
$$\int \frac{1}{\cos{\left(3 x \right)}}\, dx = C - \frac{\log{\left(\sin{\left(3 x \right)} - 1 \right)}}{6} + \frac{\log{\left(\sin{\left(3 x \right)} + 1 \right)}}{6}$$
Simplify
The graph
Plot the graph f(x)
The answer [src] 
nan
$$\text{NaN}$$ 
=
= 
nan
$$\text{NaN}$$ 
nan
Numerical answer [src] 
-4.26326539776129
-4.26326539776129
The graph
Integral of 1/cos3x dx

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

    © Mister Exam – Calculator