Mister Exam

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Identical expressions
(-y)/(two +y^ two)
( minus y) divide by (2 plus y squared )
( minus y) divide by (two plus y to the power of two)
(-y)/(2+y2)
-y/2+y2
(-y)/(2+y²)
(-y)/(2+y to the power of 2)
-y/2+y^2
(-y) divide by (2+y^2)
Similar expressions
(-y)/(2-y^2)
(y)/(2+y^2)

Integral of (-y)/(2+y^2) dx

The solution

You have entered [src]

  1          
  /          
 |           
 |   -y      
 |  ------ dy
 |       2   
 |  2 + y    
 |           
/            
0

\int\limits_{0}^{1} \frac{\left(-1\right) y}{y^{2} + 2}\, dy

Integral((-y)/(2 + y^2), (y, 0, 1))

Detail solution

We have the integral:

  /         
 |          
 |  -y      
 | ------ dy
 |      2   
 | 2 + y    
 |          
/

Rewrite the integrand

           /    2*y     \                   
           |------------|         /0\       
           | 2          |         |-|       
 -y        \y  + 0*y + 2/         \2/       
------ = - -------------- + ----------------
     2           2                     2    
2 + y                       /   ___   \     
                            |-\/ 2    |     
                            |-------*y|  + 1
                            \   2     /

  /           
 |            
 |  -y        
 | ------ dy  
 |      2    =
 | 2 + y      
 |            
/

   /                
  |                 
  |     2*y         
- | ------------ dy 
  |  2              
  | y  + 0*y + 2    
  |                 
 /                  
--------------------
         2

In the integral

   /                
  |                 
  |     2*y         
- | ------------ dy 
  |  2              
  | y  + 0*y + 2    
  |                 
 /                  
--------------------
         2

do replacement

     2
u = y

then

the integral =

   /                        
  |                         
  |   1                     
- | ----- du                
  | 2 + u                   
  |                         
 /              -log(2 + u) 
------------- = ------------
      2              2

do backward replacement

   /                                
  |                                 
  |     2*y                         
- | ------------ dy                 
  |  2                              
  | y  + 0*y + 2                    
  |                        /     2\ 
 /                     -log\2 + y / 
-------------------- = -------------
         2                   2

In the integral

do replacement

         ___ 
    -y*\/ 2  
v = ---------
        2

then

the integral =

True

do backward replacement

True

Solution is:

       /     2\
    log\2 + y /
C - -----------
         2

The answer (Indefinite) [src]

  /                           
 |                    /     2\
 |  -y             log\2 + y /
 | ------ dy = C - -----------
 |      2               2     
 | 2 + y                      
 |                            
/

\int \frac{\left(-1\right) y}{y^{2} + 2}\, dy = C - \frac{\log{\left(y^{2} + 2 \right)}}{2}

Simplify

The graph

Plot the graph f(x)

The answer [src]

log(2)   log(3)
------ - ------
  2        2

- \frac{\log{\left(3 \right)}}{2} + \frac{\log{\left(2 \right)}}{2}

log(2)   log(3)
------ - ------
  2        2

- \frac{\log{\left(3 \right)}}{2} + \frac{\log{\left(2 \right)}}{2}

log(2)/2 - log(3)/2

Numerical answer [src]

-0.202732554054082

-0.202732554054082

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 (-y)/(2+y^2) dx

Limits of integration:

The graph:

Piecewise:

The solution