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Identical expressions
one /(one thousand +6x)
1 divide by (1000 plus 6x)
one divide by (one thousand plus 6x)
1/1000+6x
1 divide by (1000+6x)
1/(1000+6x)dx
Similar expressions
1/(1000-6x)

Integral of 1/(1000+6x) dx

The solution

You have entered [src]

  1              
  /              
 |               
 |      1        
 |  ---------- dx
 |  1000 + 6*x   
 |               
/                
0

$$\int\limits_{0}^{1} \frac{1}{6 x + 1000}\, dx$$

Integral(1/(1000 + 6*x), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = 6 x + 1000$ .
  
  Then let $du = 6 dx$ and substitute $\frac{du}{6}$ :
  
  $\int \frac{1}{6 u}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{6}$
    1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
    So, the result is: $\frac{\log{\left(u \right)}}{6}$
  Now substitute $u$ back in:
  
  $\frac{\log{\left(6 x + 1000 \right)}}{6}$
Method #2
1. Rewrite the integrand:
  
  $\frac{1}{6 x + 1000} = \frac{1}{2 \left(3 x + 500\right)}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{1}{2 \left(3 x + 500\right)}\, dx = \frac{\int \frac{1}{3 x + 500}\, dx}{2}$
  1. Let $u = 3 x + 500$ .
    
    Then let $du = 3 dx$ and substitute $\frac{du}{3}$ :
    
    $\int \frac{1}{3 u}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{3}$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      So, the result is: $\frac{\log{\left(u \right)}}{3}$
    Now substitute $u$ back in:
    
    $\frac{\log{\left(3 x + 500 \right)}}{3}$
  So, the result is: $\frac{\log{\left(3 x + 500 \right)}}{6}$
Add the constant of integration:

$\frac{\log{\left(6 x + 1000 \right)}}{6}+ \mathrm{constant}$

The answer is:

$\frac{\log{\left(6 x + 1000 \right)}}{6}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                   
 |                                    
 |     1               log(1000 + 6*x)
 | ---------- dx = C + ---------------
 | 1000 + 6*x                 6       
 |                                    
/

$$\int \frac{1}{6 x + 1000}\, dx = C + \frac{\log{\left(6 x + 1000 \right)}}{6}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  log(1000)   log(1006)
- --------- + ---------
      6           6

$$- \frac{\log{\left(1000 \right)}}{6} + \frac{\log{\left(1006 \right)}}{6}$$

  log(1000)   log(1006)
- --------- + ---------
      6           6

$$- \frac{\log{\left(1000 \right)}}{6} + \frac{\log{\left(1006 \right)}}{6}$$

-log(1000)/6 + log(1006)/6

Numerical answer [src]

0.000997011946257911

0.000997011946257911

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

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Identical expressions

Similar expressions

Integral of 1/(1000+6x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2