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Solving integrals/ ((x^2)-1)/cos(x)

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

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The solution

You have entered [src] 
  1          
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 |   2       
 |  x  - 1   
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∫01x2−1cos⁡(x) dx\int\limits_{0}^{1} \frac{x^{2} - 1}{\cos{\left(x \right)}}\, dx 
Integral((x^2 - 1)/cos(x), (x, 0, 1))
Detail solution

  1. Rewrite the integrand:

    x2−1cos⁡(x)=x2cos⁡(x)−1cos⁡(x)\frac{x^{2} - 1}{\cos{\left(x \right)}} = \frac{x^{2}}{\cos{\left(x \right)}} - \frac{1}{\cos{\left(x \right)}}

  2. Integrate term-by-term:

    1. Don't know the steps in finding this integral.

      But the integral is

      ∫x2cos⁡(x) dx\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:

      ∫(−1cos⁡(x)) dx=−∫1cos⁡(x) dx\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

        −log⁡(sin⁡(x)−1)2+log⁡(sin⁡(x)+1)2- \frac{\log{\left(\sin{\left(x \right)} - 1 \right)}}{2} + \frac{\log{\left(\sin{\left(x \right)} + 1 \right)}}{2}

      So, the result is: log⁡(sin⁡(x)−1)2−log⁡(sin⁡(x)+1)2\frac{\log{\left(\sin{\left(x \right)} - 1 \right)}}{2} - \frac{\log{\left(\sin{\left(x \right)} + 1 \right)}}{2}

    The result is: log⁡(sin⁡(x)−1)2−log⁡(sin⁡(x)+1)2+∫x2cos⁡(x) dx\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

  3. Add the constant of integration:

    log⁡(sin⁡(x)−1)2−log⁡(sin⁡(x)+1)2+∫x2cos⁡(x) dx+constant\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:

log⁡(sin⁡(x)−1)2−log⁡(sin⁡(x)+1)2+∫x2cos⁡(x) dx+constant\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] 
  /                                                       /         
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 |  2                                                    |    2     
 | x  - 1          log(-1 + sin(x))   log(1 + sin(x))    |   x      
 | ------ dx = C + ---------------- - --------------- +  | ------ dx
 | cos(x)                 2                  2           | cos(x)   
 |                                                       |          
/                                                       /
∫x2−1cos⁡(x) dx=C+log⁡(sin⁡(x)−1)2−log⁡(sin⁡(x)+1)2+∫x2cos⁡(x) dx\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] 
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 |       cos(x)        
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0
∫01(x−1)(x+1)cos⁡(x) dx\int\limits_{0}^{1} \frac{\left(x - 1\right) \left(x + 1\right)}{\cos{\left(x \right)}}\, dx 
=
= 
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 |       cos(x)        
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0
∫01(x−1)(x+1)cos⁡(x) dx\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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