Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of e^(x*(-3))*dx
Integral of e^(x+e^x)
Integral of x^(2/3)*dx
Integral of e^(4*x)*dx
Identical expressions
one /(one - two *x)
1 divide by (1 minus 2 multiply by x)
one divide by (one minus two multiply by x)
1/(1-2x)
1/1-2x
1 divide by (1-2*x)
1/(1-2*x)dx
Similar expressions
1/((1-2*x)^3)
1/(1-(2x+3)^2)^1/2
1/(1-2x)^(3/2)
4/x^2+4-1/(1-2*x^2)^(13/2)
1/(1+2*x)

Integral of 1/(1-2*x) dx

The solution

You have entered [src]

 1/2          
  /           
 |            
 |     1      
 |  ------- dx
 |  1 - 2*x   
 |            
/             
0

$$\int\limits_{0}^{\frac{1}{2}} \frac{1}{1 - 2 x}\, dx$$

Integral(1/(1 - 2*x), (x, 0, 1/2))

Detail solution

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                             
 |                              
 |    1             log(1 - 2*x)
 | ------- dx = C - ------------
 | 1 - 2*x               2      
 |                              
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

     pi*I
oo + ----
      2

$$\infty + \frac{i \pi}{2}$$

     pi*I
oo + ----
      2

$$\infty + \frac{i \pi}{2}$$

oo + pi*i/2

Numerical answer [src]

22.0454783931097

22.0454783931097

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of 1/(1-2*x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3