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Identical expressions
(sqrt(ln(x+ one)))/(x+ one)
( square root of (ln(x plus 1))) divide by (x plus 1)
( square root of (ln(x plus one))) divide by (x plus one)
(√(ln(x+1)))/(x+1)
sqrtlnx+1/x+1
(sqrt(ln(x+1))) divide by (x+1)
(sqrt(ln(x+1)))/(x+1)dx
Similar expressions
(sqrt(ln(x-1)))/(x+1)
sqrt(ln(x+1))/x+1
(sqrt(ln(x+1)))/(x-1)

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

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

The solution

You have entered [src]

  1                  
  /                  
 |                   
 |    ____________   
 |  \/ log(x + 1)    
 |  -------------- dx
 |      x + 1        
 |                   
/                    
0

$$\int\limits_{0}^{1} \frac{\sqrt{\log{\left(x + 1 \right)}}}{x + 1}\, dx$$

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

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = \log{\left(x + 1 \right)}$ .
  
  Then let $du = \frac{dx}{x + 1}$ and substitute $du$ :
  
  $\int \sqrt{u}\, du$
  1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int \sqrt{u}\, du = \frac{2 u^{\frac{3}{2}}}{3}$
  Now substitute $u$ back in:
  
  $\frac{2 \log{\left(x + 1 \right)}^{\frac{3}{2}}}{3}$
Method #2
1. Let $u = x + 1$ .
  
  Then let $du = dx$ and substitute $du$ :
  
  $\int \frac{\sqrt{\log{\left(u \right)}}}{u}\, du$
  1. Let $u = \frac{1}{u}$ .
    
    Then let $du = - \frac{du}{u^{2}}$ and substitute $- du$ :
    
    $\int \left(- \frac{\sqrt{\log{\left(\frac{1}{u} \right)}}}{u}\right)\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\sqrt{\log{\left(\frac{1}{u} \right)}}}{u}\, du = - \int \frac{\sqrt{\log{\left(\frac{1}{u} \right)}}}{u}\, du$
      
      Let $u = \log{\left(\frac{1}{u} \right)}$ .
      
      Then let $du = - \frac{du}{u}$ and substitute $- du$ :
      
      $\int \left(- \sqrt{u}\right)\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \sqrt{u}\, du = - \int \sqrt{u}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \sqrt{u}\, du = \frac{2 u^{\frac{3}{2}}}{3}$
      
      So, the result is: $- \frac{2 u^{\frac{3}{2}}}{3}$
      
      Now substitute $u$ back in:
      
      $- \frac{2 \log{\left(\frac{1}{u} \right)}^{\frac{3}{2}}}{3}$
      
      So, the result is: $\frac{2 \log{\left(\frac{1}{u} \right)}^{\frac{3}{2}}}{3}$
    Now substitute $u$ back in:
    
    $\frac{2 \log{\left(u \right)}^{\frac{3}{2}}}{3}$
  Now substitute $u$ back in:
  
  $\frac{2 \log{\left(x + 1 \right)}^{\frac{3}{2}}}{3}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                                       
 |                                        
 |   ____________               3/2       
 | \/ log(x + 1)           2*log   (x + 1)
 | -------------- dx = C + ---------------
 |     x + 1                      3       
 |                                        
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

     3/2   
2*log   (2)
-----------
     3

$$\frac{2 \log{\left(2 \right)}^{\frac{3}{2}}}{3}$$

     3/2   
2*log   (2)
-----------
     3

$$\frac{2 \log{\left(2 \right)}^{\frac{3}{2}}}{3}$$

2*log(2)^(3/2)/3

Numerical answer [src]

0.384721920924093

0.384721920924093

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

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How to use it?

Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2