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Similar expressions
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Integral of ln(x)/sqrt(x^2+3) dx

The solution

You have entered [src]

 oo               
  /               
 |                
 |     log(x)     
 |  ----------- dx
 |     ________   
 |    /  2        
 |  \/  x  + 3    
 |                
/                 
1

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

Integral(log(x)/sqrt(x^2 + 3), (x, 1, oo))

Detail solution

Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(x \right)} = \log{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = \frac{1}{\sqrt{x^{2} + 3}}$ .

Then $\operatorname{du}{\left(x \right)} = \frac{1}{x}$ .

To find $v{\left(x \right)}$ :
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{1}{\sqrt{x^{2} + 3}}\, dx = \frac{\sqrt{3} \int \frac{1}{\sqrt{\frac{x^{2}}{3} + 1}}\, dx}{3}$
  1. Let $u = \frac{\sqrt{3} x}{3}$ .
    
    Then let $du = \frac{\sqrt{3} dx}{3}$ and substitute $\sqrt{3} du$ :
    
    $\int \frac{3}{\sqrt{u^{2} + 1}}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\sqrt{3}}{\sqrt{u^{2} + 1}}\, du = \sqrt{3} \int \frac{1}{\sqrt{u^{2} + 1}}\, du$
      So, the result is: $\sqrt{3} \operatorname{asinh}{\left(u \right)}$
    Now substitute $u$ back in:
    
    $\sqrt{3} \operatorname{asinh}{\left(\frac{\sqrt{3} x}{3} \right)}$
  So, the result is: $\operatorname{asinh}{\left(\frac{\sqrt{3} x}{3} \right)}$
Now evaluate the sub-integral.
Don't know the steps in finding this integral.

But the integral is

$\int \frac{\operatorname{asinh}{\left(\frac{\sqrt{3} x}{3} \right)}}{x}\, dx$
Add the constant of integration:

$\log{\left(x \right)} \operatorname{asinh}{\left(\frac{\sqrt{3} x}{3} \right)} - \int \frac{\operatorname{asinh}{\left(\frac{\sqrt{3} x}{3} \right)}}{x}\, dx+ \mathrm{constant}$

The answer is:

$\log{\left(x \right)} \operatorname{asinh}{\left(\frac{\sqrt{3} x}{3} \right)} - \int \frac{\operatorname{asinh}{\left(\frac{\sqrt{3} x}{3} \right)}}{x}\, dx+ \mathrm{constant}$

The answer (Indefinite) [src]

                          /                                         
                         |                                          
                         |      /    ___\                           
  /                      |      |x*\/ 3 |                           
 |                       | asinh|-------|           /    ___\       
 |    log(x)             |      \   3   /           |x*\/ 3 |       
 | ----------- dx = C -  | -------------- dx + asinh|-------|*log(x)
 |    ________           |       x                  \   3   /       
 |   /  2                |                                          
 | \/  x  + 3           /                                           
 |                                                                  
/

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

Simplify

The answer [src]

 oo               
  /               
 |                
 |     log(x)     
 |  ----------- dx
 |     ________   
 |    /      2    
 |  \/  3 + x     
 |                
/                 
1

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

 oo               
  /               
 |                
 |     log(x)     
 |  ----------- dx
 |     ________   
 |    /      2    
 |  \/  3 + x     
 |                
/                 
1

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

Integral(log(x)/sqrt(3 + x^2), (x, 1, oo))

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

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Identical expressions

Similar expressions

Integral of ln(x)/sqrt(x^2+3) dx

Limits of integration:

The graph:

Piecewise:

The solution