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

The solution

You have entered [src]

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

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

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

Detail solution

Rewrite the integrand:

$\frac{- \sin{\left(x \right)} + \cos{\left(x \right)}}{\sin{\left(x \right)} + 1} = - \frac{\sin{\left(x \right)}}{\sin{\left(x \right)} + 1} + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)} + 1}$
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{\sin{\left(x \right)}}{\sin{\left(x \right)} + 1}\right)\, dx = - \int \frac{\sin{\left(x \right)}}{\sin{\left(x \right)} + 1}\, dx$
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $\frac{x \tan{\left(\frac{x}{2} \right)}}{\tan{\left(\frac{x}{2} \right)} + 1} + \frac{x}{\tan{\left(\frac{x}{2} \right)} + 1} + \frac{2}{\tan{\left(\frac{x}{2} \right)} + 1}$
  So, the result is: $- \frac{x \tan{\left(\frac{x}{2} \right)}}{\tan{\left(\frac{x}{2} \right)} + 1} - \frac{x}{\tan{\left(\frac{x}{2} \right)} + 1} - \frac{2}{\tan{\left(\frac{x}{2} \right)} + 1}$
1. Let $u = \sin{\left(x \right)} + 1$ .
  
  Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
  
  $\int \frac{1}{u}\, du$
  1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
  Now substitute $u$ back in:
  
  $\log{\left(\sin{\left(x \right)} + 1 \right)}$
The result is: $- \frac{x \tan{\left(\frac{x}{2} \right)}}{\tan{\left(\frac{x}{2} \right)} + 1} - \frac{x}{\tan{\left(\frac{x}{2} \right)} + 1} + \log{\left(\sin{\left(x \right)} + 1 \right)} - \frac{2}{\tan{\left(\frac{x}{2} \right)} + 1}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

    pi         
1 - -- + log(2)
    2

$$- \frac{\pi}{2} + \log{\left(2 \right)} + 1$$

    pi         
1 - -- + log(2)
    2

$$- \frac{\pi}{2} + \log{\left(2 \right)} + 1$$

1 - pi/2 + log(2)

Numerical answer [src]

0.122350853765049

0.122350853765049

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

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Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution