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cosdx/(2sinx+1)

Integral of cosdx/(2sinx-1) dx

The solution

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

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

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

Detail solution

The integral of a constant times a function is the constant times the integral of the function:

$\int \frac{\cos{\left(1 \right)}}{2 \sin{\left(x \right)} - 1}\, dx = \cos{\left(1 \right)} \int \frac{1}{2 \sin{\left(x \right)} - 1}\, dx$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $- \frac{\sqrt{3} \log{\left(\tan{\left(\frac{x}{2} \right)} - 2 - \sqrt{3} \right)}}{3} + \frac{\sqrt{3} \log{\left(\tan{\left(\frac{x}{2} \right)} - 2 + \sqrt{3} \right)}}{3}$
So, the result is: $\left(- \frac{\sqrt{3} \log{\left(\tan{\left(\frac{x}{2} \right)} - 2 - \sqrt{3} \right)}}{3} + \frac{\sqrt{3} \log{\left(\tan{\left(\frac{x}{2} \right)} - 2 + \sqrt{3} \right)}}{3}\right) \cos{\left(1 \right)}$
Now simplify:

$\frac{\sqrt{3} \left(- \log{\left(\tan{\left(\frac{x}{2} \right)} - 2 - \sqrt{3} \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} - 2 + \sqrt{3} \right)}\right) \cos{\left(1 \right)}}{3}$
Add the constant of integration:

$\frac{\sqrt{3} \left(- \log{\left(\tan{\left(\frac{x}{2} \right)} - 2 - \sqrt{3} \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} - 2 + \sqrt{3} \right)}\right) \cos{\left(1 \right)}}{3}+ \mathrm{constant}$

The answer is:

$\frac{\sqrt{3} \left(- \log{\left(\tan{\left(\frac{x}{2} \right)} - 2 - \sqrt{3} \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} - 2 + \sqrt{3} \right)}\right) \cos{\left(1 \right)}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                      /    ___    /       ___      /x\\     ___    /       ___      /x\\\       
 |                       |  \/ 3 *log|-2 - \/ 3  + tan|-||   \/ 3 *log|-2 + \/ 3  + tan|-|||       
 |    cos(1)             |           \                \2//            \                \2//|       
 | ------------ dx = C + |- ------------------------------ + ------------------------------|*cos(1)
 | 2*sin(x) - 1          \                3                                3               /       
 |                                                                                                 
/

$$\int \frac{\cos{\left(1 \right)}}{2 \sin{\left(x \right)} - 1}\, dx = C + \left(- \frac{\sqrt{3} \log{\left(\tan{\left(\frac{x}{2} \right)} - 2 - \sqrt{3} \right)}}{3} + \frac{\sqrt{3} \log{\left(\tan{\left(\frac{x}{2} \right)} - 2 + \sqrt{3} \right)}}{3}\right) \cos{\left(1 \right)}$$

Simplify

The graph

Plot the graph f(x)

Numerical answer [src]

4.09526803133794

4.09526803133794

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

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Identical expressions

Similar expressions

Integral of cosdx/(2sinx-1) dx

Limits of integration:

The graph:

Piecewise:

The solution