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

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

The solution

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

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

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

Detail solution

Rewrite the integrand:

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

     pi*I
oo + ----
      2

$$\infty + \frac{i \pi}{2}$$

     pi*I
oo + ----
      2

$$\infty + \frac{i \pi}{2}$$

oo + pi*i/2

The graph

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

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

Limits of integration:

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Piecewise:

The solution