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Solving integrals/ log(1-x^2)/x^2

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

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src] 
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∫01log⁡(1−x2)x2 dx\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

  1. Use integration by parts:

    ∫u⁡dv⁡=u⁡v⁡−∫v⁡du⁡\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

    Let u(x)=log⁡(1−x2)u{\left(x \right)} = \log{\left(1 - x^{2} \right)} and let dv⁡(x)=1x2\operatorname{dv}{\left(x \right)} = \frac{1}{x^{2}}.

    Then du⁡(x)=−2x1−x2\operatorname{du}{\left(x \right)} = - \frac{2 x}{1 - x^{2}}.

    To find v(x)v{\left(x \right)}:

    1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n≠−1n \neq -1:

      ∫1x2 dx=−1x\int \frac{1}{x^{2}}\, dx = - \frac{1}{x}

    Now evaluate the sub-integral.

  2. The integral of a constant times a function is the constant times the integral of the function:

    ∫21−x2 dx=2∫11−x2 dx\int \frac{2}{1 - x^{2}}\, dx = 2 \int \frac{1}{1 - x^{2}}\, dx

      PiecewiseRule(subfunctions=[(ArctanRule(a=1, b=-1, c=1, context=1/(1 - x**2), symbol=x), False), (ArccothRule(a=1, b=-1, c=1, context=1/(1 - x**2), symbol=x), x**2 > 1), (ArctanhRule(a=1, b=-1, c=1, context=1/(1 - x**2), symbol=x), x**2 < 1)], context=1/(1 - x**2), symbol=x)

    So, the result is: 2({acoth⁡(x)for x2>1atanh⁡(x)for x2<1)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)

  3. Now simplify:

    {−(2acoth⁡(x)+log⁡(1−x2)x)for x2>1−(2atanh⁡(x)+log⁡(1−x2)x)for x2<1\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}

  4. Add the constant of integration:

    {−(2acoth⁡(x)+log⁡(1−x2)x)for x2>1−(2atanh⁡(x)+log⁡(1−x2)x)for x2<1+constant\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:

{−(2acoth⁡(x)+log⁡(1−x2)x)for x2>1−(2atanh⁡(x)+log⁡(1−x2)x)for x2<1+constant\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/              
 |                                                             
/
∫log⁡(1−x2)x2 dx=C−2({acoth⁡(x)for x2>1atanh⁡(x)for x2<1)−log⁡(1−x2)x\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
0.001.000.100.200.300.400.500.600.700.800.900-10
Plot the graph f(x)
The answer [src] 
-2*log(2)
−2log⁡(2)- 2 \log{\left(2 \right)} 
=
= 
-2*log(2)
−2log⁡(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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