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Similar expressions
1/(sqrt(3*x+1)+1)
1/(sqrt(3*x-1)-1)

Integral of 1/(sqrt(3*x+1)-1) dx

The solution

You have entered [src]

  1                   
  /                   
 |                    
 |         1          
 |  --------------- dx
 |    _________       
 |  \/ 3*x + 1  - 1   
 |                    
/                     
0

$$\int\limits_{0}^{1} \frac{1}{\sqrt{3 x + 1} - 1}\, dx$$

Integral(1/(sqrt(3*x + 1) - 1), (x, 0, 1))

Detail solution

Let $u = \sqrt{3 x + 1}$ .

Then let $du = \frac{3 dx}{2 \sqrt{3 x + 1}}$ and substitute $2 du$ :

$\int \frac{2 u}{3 u - 3}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{u}{3 u - 3}\, du = 2 \int \frac{u}{3 u - 3}\, du$
  1. Rewrite the integrand:
    
    $\frac{u}{3 u - 3} = \frac{1}{3} + \frac{1}{3 \left(u - 1\right)}$
  2. Integrate term-by-term:
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int \frac{1}{3}\, du = \frac{u}{3}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{3 \left(u - 1\right)}\, du = \frac{\int \frac{1}{u - 1}\, du}{3}$
      1. Let $u = u - 1$ .
        
        Then let $du = du$ and substitute $du$ :
        
        $\int \frac{1}{u}\, du$
        
        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: $\frac{\log{\left(u - 1 \right)}}{3}$
    The result is: $\frac{u}{3} + \frac{\log{\left(u - 1 \right)}}{3}$
  So, the result is: $\frac{2 u}{3} + \frac{2 \log{\left(u - 1 \right)}}{3}$
Now substitute $u$ back in:

$\frac{2 \sqrt{3 x + 1}}{3} + \frac{2 \log{\left(\sqrt{3 x + 1} - 1 \right)}}{3}$
Now simplify:

$\frac{2 \sqrt{3 x + 1}}{3} + \frac{2 \log{\left(\sqrt{3 x + 1} - 1 \right)}}{3}$
Add the constant of integration:

$\frac{2 \sqrt{3 x + 1}}{3} + \frac{2 \log{\left(\sqrt{3 x + 1} - 1 \right)}}{3}+ \mathrm{constant}$

The answer is:

$\frac{2 \sqrt{3 x + 1}}{3} + \frac{2 \log{\left(\sqrt{3 x + 1} - 1 \right)}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

oo

$$\infty$$

oo

$$\infty$$

oo

Numerical answer [src]

29.7900517291088

29.7900517291088

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

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Identical expressions

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Integral of 1/(sqrt(3*x+1)-1) dx

Limits of integration:

The graph:

Piecewise:

The solution