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Integral of 1/(1000-2y) dy

The solution

You have entered [src]

  1              
  /              
 |               
 |      1        
 |  ---------- dy
 |  1000 - 2*y   
 |               
/                
0

$$\int\limits_{0}^{1} \frac{1}{1000 - 2 y}\, dy$$

Integral(1/(1000 - 2*y), (y, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = 1000 - 2 y$ .
  
  Then let $du = - 2 dy$ and substitute $- \frac{du}{2}$ :
  
  $\int \left(- \frac{1}{2 u}\right)\, du$
  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:
  
  $- \frac{\log{\left(1000 - 2 y \right)}}{2}$
Method #2
1. Rewrite the integrand:
  
  $\frac{1}{1000 - 2 y} = - \frac{1}{2 \left(y - 500\right)}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{1}{2 \left(y - 500\right)}\right)\, dy = - \frac{\int \frac{1}{y - 500}\, dy}{2}$
  1. Let $u = y - 500$ .
    
    Then let $du = dy$ 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:
    
    $\log{\left(y - 500 \right)}$
  So, the result is: $- \frac{\log{\left(y - 500 \right)}}{2}$
Method #3
1. Rewrite the integrand:
  
  $\frac{1}{1000 - 2 y} = - \frac{1}{2 y - 1000}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{1}{2 y - 1000}\right)\, dy = - \int \frac{1}{2 y - 1000}\, dy$
  1. Let $u = 2 y - 1000$ .
    
    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:
      
      $\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{2}$
      
      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:
    
    $\frac{\log{\left(2 y - 1000 \right)}}{2}$
  So, the result is: $- \frac{\log{\left(2 y - 1000 \right)}}{2}$
Add the constant of integration:

$- \frac{\log{\left(1000 - 2 y \right)}}{2}+ \mathrm{constant}$

The answer is:

$- \frac{\log{\left(1000 - 2 y \right)}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                   
 |                                    
 |     1               log(1000 - 2*y)
 | ---------- dy = C - ---------------
 | 1000 - 2*y                 2       
 |                                    
/

$$\int \frac{1}{1000 - 2 y}\, dy = C - \frac{\log{\left(1000 - 2 y \right)}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

log(1000)   log(998)
--------- - --------
    2          2

$$- \frac{\log{\left(998 \right)}}{2} + \frac{\log{\left(1000 \right)}}{2}$$

log(1000)   log(998)
--------- - --------
    2          2

$$- \frac{\log{\left(998 \right)}}{2} + \frac{\log{\left(1000 \right)}}{2}$$

log(1000)/2 - log(998)/2

Numerical answer [src]

0.00100100133533654

0.00100100133533654

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

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Identical expressions

Similar expressions

Integral of 1/(1000-2y) dy

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3