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log(1-x^2)/x^2
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Similar expressions
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Integral of log(1-x^2)/x^2 dx

The solution

You have entered [src]

  1               
  /               
 |                
 |     /     2\   
 |  log\1 - x /   
 |  ----------- dx
 |        2       
 |       x        
 |                
/                 
0

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

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

Detail solution

Use integration by parts:

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

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

Then $\operatorname{du}{\left(x \right)} = - \frac{2 x}{1 - x^{2}}$ .

To find $v{\left(x \right)}$ :
1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int \frac{1}{x^{2}}\, dx = - \frac{1}{x}$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int \frac{2}{1 - x^{2}}\, dx = 2 \int \frac{1}{1 - x^{2}}\, dx$
So, the result is: $2 \left(\begin{cases} \operatorname{acoth}{\left(x \right)} & \text{for}\: x^{2} > 1 \\\operatorname{atanh}{\left(x \right)} & \text{for}\: x^{2} < 1 \end{cases}\right)$
Now simplify:

$\begin{cases} - (2 \operatorname{acoth}{\left(x \right)} + \frac{\log{\left(1 - x^{2} \right)}}{x}) & \text{for}\: x^{2} > 1 \\- (2 \operatorname{atanh}{\left(x \right)} + \frac{\log{\left(1 - x^{2} \right)}}{x}) & \text{for}\: x^{2} < 1 \end{cases}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                                                            
 |                                                             
 |    /     2\            //               2    \      /     2\
 | log\1 - x /            ||acoth(x)  for x  > 1|   log\1 - x /
 | ----------- dx = C - 2*|<                    | - -----------
 |       2                ||               2    |        x     
 |      x                 \\atanh(x)  for x  < 1/              
 |                                                             
/

$$\int \frac{\log{\left(1 - x^{2} \right)}}{x^{2}}\, dx = C - 2 \left(\begin{cases} \operatorname{acoth}{\left(x \right)} & \text{for}\: x^{2} > 1 \\\operatorname{atanh}{\left(x \right)} & \text{for}\: x^{2} < 1 \end{cases}\right) - \frac{\log{\left(1 - x^{2} \right)}}{x}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-2*log(2)

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

-2*log(2)

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

-2*log(2)

Numerical answer [src]

-1.38629436111302

-1.38629436111302

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

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Identical expressions

Similar expressions

Integral of log(1-x^2)/x^2 dx

Limits of integration:

The graph:

Piecewise:

The solution