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Solving integrals/ 1/(x^(1/2)+1)

Integral of 1/(x^(1/2)+1) dx

The solution

You have entered [src]

  1             
  /             
 |              
 |      1       
 |  --------- dx
 |    ___       
 |  \/ x  + 1   
 |              
/               
0

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

Integral(1/(sqrt(x) + 1), (x, 0, 1))

Detail solution

Let $u = \sqrt{x}$ .

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

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

2 - 2*log(2)

2 - 2 \log{\left(2 \right)}

2 - 2*log(2)

2 - 2 \log{\left(2 \right)}

2 - 2*log(2)

Numerical answer [src]

0.613705638880109

0.613705638880109

The graph

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

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Integral of 1/(x^(1/2)+1) dx

Limits of integration:

The graph:

Piecewise:

The solution