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Solving integrals/ sinx/(4-cos^2x)

Integral of sinx/(4-cos^2x) dx

The solution

You have entered [src]

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

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

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

Detail solution

Rewrite the integrand:

$\frac{\sin{\left(x \right)}}{4 - \cos^{2}{\left(x \right)}} = - \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)} - 4}$
The integral of a constant times a function is the constant times the integral of the function:

$\int \left(- \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)} - 4}\right)\, dx = - \int \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)} - 4}\, dx$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $- \frac{\log{\left(\cos{\left(x \right)} - 2 \right)}}{4} + \frac{\log{\left(\cos{\left(x \right)} + 2 \right)}}{4}$
So, the result is: $\frac{\log{\left(\cos{\left(x \right)} - 2 \right)}}{4} - \frac{\log{\left(\cos{\left(x \right)} + 2 \right)}}{4}$
Add the constant of integration:

$\frac{\log{\left(\cos{\left(x \right)} - 2 \right)}}{4} - \frac{\log{\left(\cos{\left(x \right)} + 2 \right)}}{4}+ \mathrm{constant}$

The answer is:

$\frac{\log{\left(\cos{\left(x \right)} - 2 \right)}}{4} - \frac{\log{\left(\cos{\left(x \right)} + 2 \right)}}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                       
 |                                                        
 |    sin(x)            log(2 + cos(x))   log(-2 + cos(x))
 | ----------- dx = C - --------------- + ----------------
 |        2                    4                 4        
 | 4 - cos (x)                                            
 |                                                        
/

$$\int \frac{\sin{\left(x \right)}}{4 - \cos^{2}{\left(x \right)}}\, dx = C + \frac{\log{\left(\cos{\left(x \right)} - 2 \right)}}{4} - \frac{\log{\left(\cos{\left(x \right)} + 2 \right)}}{4}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  log(2 + cos(1))   log(3)   log(2 - cos(1))
- --------------- + ------ + ---------------
         4            4             4

$$- \frac{\log{\left(\cos{\left(1 \right)} + 2 \right)}}{4} + \frac{\log{\left(2 - \cos{\left(1 \right)} \right)}}{4} + \frac{\log{\left(3 \right)}}{4}$$

  log(2 + cos(1))   log(3)   log(2 - cos(1))
- --------------- + ------ + ---------------
         4            4             4

$$- \frac{\log{\left(\cos{\left(1 \right)} + 2 \right)}}{4} + \frac{\log{\left(2 - \cos{\left(1 \right)} \right)}}{4} + \frac{\log{\left(3 \right)}}{4}$$

-log(2 + cos(1))/4 + log(3)/4 + log(2 - cos(1))/4

Numerical answer [src]

0.136139638032061

0.136139638032061

The graph

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

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Integral of sinx/(4-cos^2x) dx

Limits of integration:

The graph:

Piecewise:

The solution