Integral of cos2x/cosx dx

The solution

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  1            
  /            
 |             
 |  cos(2*x)   
 |  -------- dx
 |   cos(x)    
 |             
/              
0

$$\int\limits_{0}^{1} \frac{\cos{\left(2 x \right)}}{\cos{\left(x \right)}}\, dx$$

Integral(cos(2*x)/cos(x), (x, 0, 1))

Detail solution

Rewrite the integrand:

$\frac{\cos{\left(2 x \right)}}{\cos{\left(x \right)}} = 2 \cos{\left(x \right)} - \frac{1}{\cos{\left(x \right)}}$
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 2 \cos{\left(x \right)}\, dx = 2 \int \cos{\left(x \right)}\, dx$
  1. The integral of cosine is sine:
    
    $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
  So, the result is: $2 \sin{\left(x \right)}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{1}{\cos{\left(x \right)}}\right)\, dx = - \int \frac{1}{\cos{\left(x \right)}}\, dx$
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $- \frac{\log{\left(\sin{\left(x \right)} - 1 \right)}}{2} + \frac{\log{\left(\sin{\left(x \right)} + 1 \right)}}{2}$
  So, the result is: $\frac{\log{\left(\sin{\left(x \right)} - 1 \right)}}{2} - \frac{\log{\left(\sin{\left(x \right)} + 1 \right)}}{2}$
The result is: $\frac{\log{\left(\sin{\left(x \right)} - 1 \right)}}{2} - \frac{\log{\left(\sin{\left(x \right)} + 1 \right)}}{2} + 2 \sin{\left(x \right)}$
Add the constant of integration:

$\frac{\log{\left(\sin{\left(x \right)} - 1 \right)}}{2} - \frac{\log{\left(\sin{\left(x \right)} + 1 \right)}}{2} + 2 \sin{\left(x \right)}+ \mathrm{constant}$

The answer is:

$\frac{\log{\left(\sin{\left(x \right)} - 1 \right)}}{2} - \frac{\log{\left(\sin{\left(x \right)} + 1 \right)}}{2} + 2 \sin{\left(x \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                               
 |                                                                
 | cos(2*x)          log(-1 + sin(x))              log(1 + sin(x))
 | -------- dx = C + ---------------- + 2*sin(x) - ---------------
 |  cos(x)                  2                             2       
 |                                                                
/

$$\int \frac{\cos{\left(2 x \right)}}{\cos{\left(x \right)}}\, dx = C + \frac{\log{\left(\sin{\left(x \right)} - 1 \right)}}{2} - \frac{\log{\left(\sin{\left(x \right)} + 1 \right)}}{2} + 2 \sin{\left(x \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

log(1 - sin(1))              log(1 + sin(1))
--------------- + 2*sin(1) - ---------------
       2                            2

$$\frac{\log{\left(1 - \sin{\left(1 \right)} \right)}}{2} - \frac{\log{\left(\sin{\left(1 \right)} + 1 \right)}}{2} + 2 \sin{\left(1 \right)}$$

log(1 - sin(1))              log(1 + sin(1))
--------------- + 2*sin(1) - ---------------
       2                            2

$$\frac{\log{\left(1 - \sin{\left(1 \right)} \right)}}{2} - \frac{\log{\left(\sin{\left(1 \right)} + 1 \right)}}{2} + 2 \sin{\left(1 \right)}$$

log(1 - sin(1))/2 + 2*sin(1) - log(1 + sin(1))/2

Numerical answer [src]

0.456750798732276

0.456750798732276

The graph

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

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Identical expressions

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Integral of cos2x/cosx dx

Limits of integration:

The graph:

Piecewise:

The solution