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Integral of 1/(2n-sqrt(n)) dx

The solution

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 oo               
  /               
 |                
 |       1        
 |  ----------- dn
 |          ___   
 |  2*n - \/ n    
 |                
/                 
1

$$\int\limits_{1}^{\infty} \frac{1}{- \sqrt{n} + 2 n}\, dn$$

Integral(1/(2*n - sqrt(n)), (n, 1, oo))

Detail solution

Let $u = \sqrt{n}$ .

Then let $du = \frac{dn}{2 \sqrt{n}}$ and substitute $2 du$ :

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

$\log{\left(2 \sqrt{n} - 1 \right)}$
Add the constant of integration:

$\log{\left(2 \sqrt{n} - 1 \right)}+ \mathrm{constant}$

The answer is:

$\log{\left(2 \sqrt{n} - 1 \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                      
 |                                       
 |      1                  /         ___\
 | ----------- dn = C + log\-1 + 2*\/ n /
 |         ___                           
 | 2*n - \/ n                            
 |                                       
/

$$\int \frac{1}{- \sqrt{n} + 2 n}\, dn = C + \log{\left(2 \sqrt{n} - 1 \right)}$$

Simplify

The graph

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The answer [src]

oo

$$\infty$$

oo

$$\infty$$

oo

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

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Identical expressions

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Integral of 1/(2n-sqrt(n)) dx

Limits of integration:

The graph:

Piecewise:

The solution