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

The solution

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  0               
  /               
 |                
 |       1        
 |  ----------- dx
 |         2      
 |  1 - cos (x)   
 |                
/                 
pi                
--                
4

$$\int\limits_{\frac{\pi}{4}}^{0} \frac{1}{1 - \cos^{2}{\left(x \right)}}\, dx$$

Integral(1/(1 - cos(x)^2), (x, pi/4, 0))

Detail solution

Rewrite the integrand:

$\frac{1}{1 - \cos^{2}{\left(x \right)}} = - \frac{1}{\cos^{2}{\left(x \right)} - 1}$
The integral of a constant times a function is the constant times the integral of the function:

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                        /x\           
 |                      tan|-|           
 |      1                  \2/      1    
 | ----------- dx = C + ------ - --------
 |        2               2           /x\
 | 1 - cos (x)                   2*tan|-|
 |                                    \2/
/

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

Simplify

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Plot the graph f(x)

The answer [src]

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$$-\infty$$

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$$-\infty$$

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

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

Limits of integration:

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Piecewise:

The solution