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Integral of ((x^2)-1)/cos(x) dx

The solution

You have entered [src]

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

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

Integral((x^2 - 1)/cos(x), (x, 0, 1))

Detail solution

Rewrite the integrand:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The answer [src]

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

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

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

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

Integral((1 + x)*(-1 + x)/cos(x), (x, 0, 1))

Numerical answer [src]

-0.748900737919218

-0.748900737919218

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

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Identical expressions

Similar expressions

Integral of ((x^2)-1)/cos(x) dx

Limits of integration:

The graph:

Piecewise:

The solution