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Solving integrals/ 1/(1+sqrt(x+1))

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

The solution

You have entered [src]

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

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

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

Detail solution

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

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

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

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

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

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

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

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

Numerical answer [src]

0.451974311826995

0.451974311826995

The graph

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

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Identical expressions

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

Limits of integration:

The graph:

Piecewise:

The solution