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Similar expressions
(dx)/(1+sqrt(2x+1))
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Integral of (dx)/(1+sqrt(2x-1)) dx

The solution

You have entered [src]

  3                   
  /                   
 |                    
 |         1          
 |  --------------- dx
 |        _________   
 |  1 + \/ 2*x - 1    
 |                    
/                     
1

$$\int\limits_{1}^{3} \frac{1}{\sqrt{2 x - 1} + 1}\, dx$$

Integral(1/(1 + sqrt(2*x - 1)), (x, 1, 3))

Detail solution

Let $u = \sqrt{2 x - 1}$ .

Then let $du = \frac{dx}{\sqrt{2 x - 1}}$ and substitute $du$ :

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

$\sqrt{2 x - 1} - \log{\left(\sqrt{2 x - 1} + 1 \right)}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                                                           
 |                                                            
 |        1                   _________      /      _________\
 | --------------- dx = C + \/ 2*x - 1  - log\1 + \/ 2*x - 1 /
 |       _________                                            
 | 1 + \/ 2*x - 1                                             
 |                                                            
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

       ___      /      ___\         
-1 + \/ 5  - log\1 + \/ 5 / + log(2)

$$- \log{\left(1 + \sqrt{5} \right)} - 1 + \log{\left(2 \right)} + \sqrt{5}$$

       ___      /      ___\         
-1 + \/ 5  - log\1 + \/ 5 / + log(2)

$$- \log{\left(1 + \sqrt{5} \right)} - 1 + \log{\left(2 \right)} + \sqrt{5}$$

-1 + sqrt(5) - log(1 + sqrt(5)) + log(2)

Numerical answer [src]

0.754856152440186

0.754856152440186

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

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Identical expressions

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Integral of (dx)/(1+sqrt(2x-1)) dx

Limits of integration:

The graph:

Piecewise:

The solution