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Similar expressions
xlnx/((sqrt(x^2+1)^3))

Solving integrals/ xlnx/((sqrt(x^2-1)^3))

Integral of xlnx/((sqrt(x^2-1)^3)) dx

The solution

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  1                
  /                
 |                 
 |    x*log(x)     
 |  ------------ dx
 |             3   
 |     ________    
 |    /  2         
 |  \/  x  - 1     
 |                 
/                  
0

$$\int\limits_{0}^{1} \frac{x \log{\left(x \right)}}{\left(\sqrt{x^{2} - 1}\right)^{3}}\, dx$$

Integral(x*log(x)/((sqrt(x^2 - 1*1))^3), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\frac{x \log{\left(x \right)}}{\left(\sqrt{x^{2} - 1}\right)^{3}} = \frac{x \log{\left(x \right)}}{x^{2} \sqrt{x^{2} - 1} - \sqrt{x^{2} - 1}}$
2. 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 $\operatorname{dv}{\left(x \right)} = \frac{x}{x^{2} \sqrt{x^{2} - 1} - \sqrt{x^{2} - 1}}$ .
  
  Then $\operatorname{du}{\left(x \right)} = \frac{1}{x}$ .
  
  To find $v{\left(x \right)}$ :
  1. Let $u = x^{2} - 1$ .
    
    Then let $du = 2 x dx$ and substitute $\frac{du}{2}$ :
    
    $\int \frac{1}{4 u^{\frac{3}{2}}}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{2 u^{\frac{3}{2}}}\, du = \frac{\int \frac{1}{u^{\frac{3}{2}}}\, du}{2}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{\frac{3}{2}}}\, du = - \frac{2}{\sqrt{u}}$
      
      So, the result is: $- \frac{1}{\sqrt{u}}$
    Now substitute $u$ back in:
    
    $- \frac{1}{\sqrt{x^{2} - 1}}$
  Now evaluate the sub-integral.
3. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{1}{x \sqrt{x^{2} - 1}}\right)\, dx = - \int \frac{1}{x \sqrt{x^{2} - 1}}\, dx$
  So, the result is: $- \begin{cases} \operatorname{acos}{\left(\frac{1}{x} \right)} & \text{for}\: x > -1 \wedge x < 1 \end{cases}$
Method #2
1. 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 $\operatorname{dv}{\left(x \right)} = \frac{x}{\left(x^{2} - 1\right)^{\frac{3}{2}}}$ .
  
  Then $\operatorname{du}{\left(x \right)} = \frac{1}{x}$ .
  
  To find $v{\left(x \right)}$ :
  1. Let $u = x^{2} - 1$ .
    
    Then let $du = 2 x dx$ and substitute $\frac{du}{2}$ :
    
    $\int \frac{1}{4 u^{\frac{3}{2}}}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{2 u^{\frac{3}{2}}}\, du = \frac{\int \frac{1}{u^{\frac{3}{2}}}\, du}{2}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{\frac{3}{2}}}\, du = - \frac{2}{\sqrt{u}}$
      
      So, the result is: $- \frac{1}{\sqrt{u}}$
    Now substitute $u$ back in:
    
    $- \frac{1}{\sqrt{x^{2} - 1}}$
  Now evaluate the sub-integral.
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{1}{x \sqrt{x^{2} - 1}}\right)\, dx = - \int \frac{1}{x \sqrt{x^{2} - 1}}\, dx$
  So, the result is: $- \begin{cases} \operatorname{acos}{\left(\frac{1}{x} \right)} & \text{for}\: x > -1 \wedge x < 1 \end{cases}$
Method #3
1. Rewrite the integrand:
  
  $\frac{x \log{\left(x \right)}}{\left(\sqrt{x^{2} - 1}\right)^{3}} = \frac{x \log{\left(x \right)}}{x^{2} \sqrt{x^{2} - 1} - \sqrt{x^{2} - 1}}$
2. 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 $\operatorname{dv}{\left(x \right)} = \frac{x}{x^{2} \sqrt{x^{2} - 1} - \sqrt{x^{2} - 1}}$ .
  
  Then $\operatorname{du}{\left(x \right)} = \frac{1}{x}$ .
  
  To find $v{\left(x \right)}$ :
  1. Let $u = x^{2} - 1$ .
    
    Then let $du = 2 x dx$ and substitute $\frac{du}{2}$ :
    
    $\int \frac{1}{4 u^{\frac{3}{2}}}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{2 u^{\frac{3}{2}}}\, du = \frac{\int \frac{1}{u^{\frac{3}{2}}}\, du}{2}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{\frac{3}{2}}}\, du = - \frac{2}{\sqrt{u}}$
      
      So, the result is: $- \frac{1}{\sqrt{u}}$
    Now substitute $u$ back in:
    
    $- \frac{1}{\sqrt{x^{2} - 1}}$
  Now evaluate the sub-integral.
3. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{1}{x \sqrt{x^{2} - 1}}\right)\, dx = - \int \frac{1}{x \sqrt{x^{2} - 1}}\, dx$
  So, the result is: $- \begin{cases} \operatorname{acos}{\left(\frac{1}{x} \right)} & \text{for}\: x > -1 \wedge x < 1 \end{cases}$
Now simplify:

$\begin{cases} \operatorname{acos}{\left(\frac{1}{x} \right)} - \frac{\log{\left(x \right)}}{\sqrt{x^{2} - 1}} & \text{for}\: x > -1 \wedge x < 1 \\\text{NaN} & \text{otherwise} \end{cases}$
Add the constant of integration:

$\begin{cases} \operatorname{acos}{\left(\frac{1}{x} \right)} - \frac{\log{\left(x \right)}}{\sqrt{x^{2} - 1}} & \text{for}\: x > -1 \wedge x < 1 \\\text{NaN} & \text{otherwise} \end{cases}+ \mathrm{constant}$

The answer is:

$\begin{cases} \operatorname{acos}{\left(\frac{1}{x} \right)} - \frac{\log{\left(x \right)}}{\sqrt{x^{2} - 1}} & \text{for}\: x > -1 \wedge x < 1 \\\text{NaN} & \text{otherwise} \end{cases}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

$$-\arcsin \left({{1}\over{\left| x\right| }}\right)-{{\log x}\over{ \sqrt{x^2-1}}}$$

Simplify

The answer [src]

-I*log(2)

$$-{{\pi}\over{2}}$$

-I*log(2)

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

Simplify

Numerical answer [src]

(0.0 - 0.693147180323136j)

(0.0 - 0.693147180323136j)

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 xlnx/((sqrt(x^2-1)^3)) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3