Mister Exam

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1/(1-2x)^(3/2)dx
Similar expressions
1/(1+2x)^(3/2)

Integral of 1/(1-2x)^(3/2) dx

The solution

You have entered [src]

 -1                
  /                
 |                 
 |       1         
 |  ------------ dx
 |           3/2   
 |  (1 - 2*x)      
 |                 
/                  
-2

$$\int\limits_{-2}^{-1} \frac{1}{\left(1 - 2 x\right)^{\frac{3}{2}}}\, dx$$

Integral(1/((1 - 2*x)^(3/2)), (x, -2, -1))

Detail solution

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

$\frac{1}{\sqrt{1 - 2 x}}+ \mathrm{constant}$

The answer is:

$\frac{1}{\sqrt{1 - 2 x}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                 
 |                                  
 |      1                     1     
 | ------------ dx = C + -----------
 |          3/2            _________
 | (1 - 2*x)             \/ 1 - 2*x 
 |                                  
/

$$\int \frac{1}{\left(1 - 2 x\right)^{\frac{3}{2}}}\, dx = C + \frac{1}{\sqrt{1 - 2 x}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

    ___     ___
  \/ 5    \/ 3 
- ----- + -----
    5       3

$$- \frac{\sqrt{5}}{5} + \frac{\sqrt{3}}{3}$$

    ___     ___
  \/ 5    \/ 3 
- ----- + -----
    5       3

$$- \frac{\sqrt{5}}{5} + \frac{\sqrt{3}}{3}$$

-sqrt(5)/5 + sqrt(3)/3

Numerical answer [src]

0.130136673689668

0.130136673689668

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

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Identical expressions

Similar expressions

Integral of 1/(1-2x)^(3/2) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2