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Graphing y =:
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Solving integrals/ 1/(sin(x)-cos(x))

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

The solution

You have entered [src]

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

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

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

Detail solution

Rewrite the integrand:

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

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

Numerical answer [src]

1.02684047752354

1.02684047752354

The graph

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

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Identical expressions

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

Limits of integration:

The graph:

Piecewise:

The solution