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Identical expressions
(- one)/(two *y+ one)
( minus 1) divide by (2 multiply by y plus 1)
( minus one) divide by (two multiply by y plus one)
(-1)/(2y+1)
-1/2y+1
(-1) divide by (2*y+1)
Similar expressions
(-1)/(2*y-1)
(1)/(2*y+1)

Integral of (-1)/(2*y+1) dx

The solution

You have entered [src]

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

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

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

Detail solution

The integral of a constant times a function is the constant times the integral of the function:

$\int \left(- \frac{1}{2 y + 1}\right)\, dy = - \int \frac{1}{2 y + 1}\, dy$
1. Let $u = 2 y + 1$ .
  
  Then let $du = 2 dy$ and substitute $\frac{du}{2}$ :
  
  $\int \frac{1}{2 u}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{2}$
    1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
    So, the result is: $\frac{\log{\left(u \right)}}{2}$
  Now substitute $u$ back in:
  
  $\frac{\log{\left(2 y + 1 \right)}}{2}$
So, the result is: $- \frac{\log{\left(2 y + 1 \right)}}{2}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                             
 |                              
 |   -1             log(2*y + 1)
 | ------- dy = C - ------------
 | 2*y + 1               2      
 |                              
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

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

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

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

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

-log(3)/2

Numerical answer [src]

-0.549306144334055

-0.549306144334055

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

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Identical expressions

Similar expressions

Integral of (-1)/(2*y+1) dx

Limits of integration:

The graph:

Piecewise:

The solution