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

The solution

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

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

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

Detail solution

Rewrite the integrand:

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

$\int \left(- \frac{1}{- 2 \sin{\left(x \right)} + \cos{\left(x \right)}}\right)\, dx = - \int \frac{1}{- 2 \sin{\left(x \right)} + \cos{\left(x \right)}}\, dx$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $\frac{\sqrt{5} \log{\left(\tan{\left(\frac{x}{2} \right)} + 2 + \sqrt{5} \right)}}{5} - \frac{\sqrt{5} \log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{5} + 2 \right)}}{5}$
So, the result is: $- \frac{\sqrt{5} \log{\left(\tan{\left(\frac{x}{2} \right)} + 2 + \sqrt{5} \right)}}{5} + \frac{\sqrt{5} \log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{5} + 2 \right)}}{5}$
Now simplify:

$\frac{\sqrt{5} \left(- \log{\left(\tan{\left(\frac{x}{2} \right)} + 2 + \sqrt{5} \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{5} + 2 \right)}\right)}{5}$
Add the constant of integration:

$\frac{\sqrt{5} \left(- \log{\left(\tan{\left(\frac{x}{2} \right)} + 2 + \sqrt{5} \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{5} + 2 \right)}\right)}{5}+ \mathrm{constant}$

The answer is:

$\frac{\sqrt{5} \left(- \log{\left(\tan{\left(\frac{x}{2} \right)} + 2 + \sqrt{5} \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{5} + 2 \right)}\right)}{5}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                             ___    /      ___      /x\\     ___    /      ___      /x\\
 |                            \/ 5 *log|2 + \/ 5  + tan|-||   \/ 5 *log|2 - \/ 5  + tan|-||
 |         1                           \               \2//            \               \2//
 | ----------------- dx = C - ----------------------------- + -----------------------------
 | 2*sin(x) - cos(x)                        5                               5              
 |                                                                                         
/

$$\int \frac{1}{2 \sin{\left(x \right)} - \cos{\left(x \right)}}\, dx = C - \frac{\sqrt{5} \log{\left(\tan{\left(\frac{x}{2} \right)} + 2 + \sqrt{5} \right)}}{5} + \frac{\sqrt{5} \log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{5} + 2 \right)}}{5}$$

Simplify

The graph

Plot the graph f(x)

Numerical answer [src]

-13.2057095450934

-13.2057095450934

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

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Identical expressions

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

Limits of integration:

The graph:

Piecewise:

The solution