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x/((x^2+1)^(3/2)*ln2)

Solving integrals/ x/((x^2-1)^(3/2)*ln2)

Integral of x/((x^2-1)^(3/2)*ln2) dx

The solution

You have entered [src]

  2                      
  /                      
 |                       
 |          x            
 |  ------------------ dx
 |          3/2          
 |  / 2    \             
 |  \x  - 1/   *log(2)   
 |                       
/                        
1

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

Integral(x/(((x^2 - 1)^(3/2)*log(2))), (x, 1, 2))

Detail solution

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

oo

$$\infty$$

oo

$$\infty$$

oo

Numerical answer [src]

3808486227.45872

3808486227.45872

The graph

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

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Identical expressions

Similar expressions

Integral of x/((x^2-1)^(3/2)*ln2) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2