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Integral of (1-sinx)/(cosx+sinx) dx

The solution

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

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

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

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\frac{1 - \sin{\left(x \right)}}{\sin{\left(x \right)} + \cos{\left(x \right)}} = - \frac{\sin{\left(x \right)} - 1}{\sin{\left(x \right)} + \cos{\left(x \right)}}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{\sin{\left(x \right)} - 1}{\sin{\left(x \right)} + \cos{\left(x \right)}}\right)\, dx = - \int \frac{\sin{\left(x \right)} - 1}{\sin{\left(x \right)} + \cos{\left(x \right)}}\, dx$
  1. Rewrite the integrand:
    
    $\frac{\sin{\left(x \right)} - 1}{\sin{\left(x \right)} + \cos{\left(x \right)}} = \frac{\sin{\left(x \right)}}{\sin{\left(x \right)} + \cos{\left(x \right)}} - \frac{1}{\sin{\left(x \right)} + \cos{\left(x \right)}}$
  2. Integrate term-by-term:
    1. Don't know the steps in finding this integral.
      
      But the integral is
      
      $\frac{x}{2} - \frac{\log{\left(\sin{\left(x \right)} + \cos{\left(x \right)} \right)}}{2}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{\sin{\left(x \right)} + \cos{\left(x \right)}}\right)\, dx = - \int \frac{1}{\sin{\left(x \right)} + \cos{\left(x \right)}}\, dx$
      
      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: $- \frac{\sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 + \sqrt{2} \right)}}{2} + \frac{\sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{2} - 1 \right)}}{2}$
    The result is: $\frac{x}{2} - \frac{\log{\left(\sin{\left(x \right)} + \cos{\left(x \right)} \right)}}{2} - \frac{\sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 + \sqrt{2} \right)}}{2} + \frac{\sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{2} - 1 \right)}}{2}$
  So, the result is: $- \frac{x}{2} + \frac{\log{\left(\sin{\left(x \right)} + \cos{\left(x \right)} \right)}}{2} + \frac{\sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 + \sqrt{2} \right)}}{2} - \frac{\sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{2} - 1 \right)}}{2}$
Method #2
1. Rewrite the integrand:
  
  $\frac{1 - \sin{\left(x \right)}}{\sin{\left(x \right)} + \cos{\left(x \right)}} = - \frac{\sin{\left(x \right)}}{\sin{\left(x \right)} + \cos{\left(x \right)}} + \frac{1}{\sin{\left(x \right)} + \cos{\left(x \right)}}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{\sin{\left(x \right)}}{\sin{\left(x \right)} + \cos{\left(x \right)}}\right)\, dx = - \int \frac{\sin{\left(x \right)}}{\sin{\left(x \right)} + \cos{\left(x \right)}}\, dx$
    1. Don't know the steps in finding this integral.
      
      But the integral is
      
      $\frac{x}{2} - \frac{\log{\left(\sin{\left(x \right)} + \cos{\left(x \right)} \right)}}{2}$
    So, the result is: $- \frac{x}{2} + \frac{\log{\left(\sin{\left(x \right)} + \cos{\left(x \right)} \right)}}{2}$
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $\frac{\sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 + \sqrt{2} \right)}}{2} - \frac{\sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{2} - 1 \right)}}{2}$
  The result is: $- \frac{x}{2} + \frac{\log{\left(\sin{\left(x \right)} + \cos{\left(x \right)} \right)}}{2} + \frac{\sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 + \sqrt{2} \right)}}{2} - \frac{\sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{2} - 1 \right)}}{2}$
Now simplify:

$- \frac{x}{2} + \frac{\sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 + \sqrt{2} \right)}}{2} - \frac{\sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{2} - 1 \right)}}{2} + \frac{\log{\left(\sin{\left(x + \frac{\pi}{4} \right)} \right)}}{2} + \frac{\log{\left(2 \right)}}{4}$
Add the constant of integration:

$- \frac{x}{2} + \frac{\sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 + \sqrt{2} \right)}}{2} - \frac{\sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{2} - 1 \right)}}{2} + \frac{\log{\left(\sin{\left(x + \frac{\pi}{4} \right)} \right)}}{2} + \frac{\log{\left(2 \right)}}{4}+ \mathrm{constant}$

The answer is:

$- \frac{x}{2} + \frac{\sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 + \sqrt{2} \right)}}{2} - \frac{\sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{2} - 1 \right)}}{2} + \frac{\log{\left(\sin{\left(x + \frac{\pi}{4} \right)} \right)}}{2} + \frac{\log{\left(2 \right)}}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

$$0$$

Numerical answer [src]

0.0

0.0

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

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How to use it?

Identical expressions

Similar expressions

Integral of (1-sinx)/(cosx+sinx) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2