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x*ln((1-x)/(1+x))

Integral of x*ln((1-x)/(1+x)) dx

Limits of integration:

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The graph:

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Piecewise:

The solution

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  1                
  /                
 |                 
 |       /1 - x\   
 |  x*log|-----| dx
 |       \1 + x/   
 |                 
/                  
0                  
$$\int\limits_{0}^{1} x \log{\left(\frac{1 - x}{x + 1} \right)}\, dx$$
Integral(x*log((1 - x)/(1 + x)), (x, 0, 1))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Rewrite the integrand:

    2. Use integration by parts:

      Let and let .

      Then .

      To find :

      1. The integral of is when :

      Now evaluate the sub-integral.

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

      1. Rewrite the integrand:

      2. Integrate term-by-term:

        1. The integral of a constant is the constant times the variable of integration:

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

          1. Let .

            Then let and substitute :

            1. The integral of is .

            Now substitute back in:

          So, the result is:

        1. Let .

          Then let and substitute :

          1. The integral of is .

          Now substitute back in:

        The result is:

      So, the result is:

    Method #2

    1. Use integration by parts:

      Let and let .

      Then .

      To find :

      1. The integral of is when :

      Now evaluate the sub-integral.

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

      1. Let .

        Then let and substitute :

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

          1. Rewrite the integrand:

          2. Integrate term-by-term:

            1. The integral of a constant is the constant times the variable of integration:

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

              1. Let .

                Then let and substitute :

                1. The integral of is .

                Now substitute back in:

              So, the result is:

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

              1. Let .

                Then let and substitute :

                1. The integral of is .

                Now substitute back in:

              So, the result is:

            The result is:

          So, the result is:

        Now substitute back in:

      So, the result is:

    Method #3

    1. Rewrite the integrand:

    2. Use integration by parts:

      Let and let .

      Then .

      To find :

      1. The integral of is when :

      Now evaluate the sub-integral.

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

      1. Rewrite the integrand:

      2. Integrate term-by-term:

        1. The integral of a constant is the constant times the variable of integration:

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

          1. Let .

            Then let and substitute :

            1. The integral of is .

            Now substitute back in:

          So, the result is:

        1. Let .

          Then let and substitute :

          1. The integral of is .

          Now substitute back in:

        The result is:

      So, the result is:

  2. Now simplify:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                      2    /  1       x  \
 |                                                      x *log|----- - -----|
 |      /1 - x\          log(1 + x)       log(-1 + x)         \1 + x   1 + x/
 | x*log|-----| dx = C + ---------- - x - ----------- + ---------------------
 |      \1 + x/              2                 2                  2          
 |                                                                           
/                                                                            
$$\int x \log{\left(\frac{1 - x}{x + 1} \right)}\, dx = C + \frac{x^{2} \log{\left(- \frac{x}{x + 1} + \frac{1}{x + 1} \right)}}{2} - x - \frac{\log{\left(x - 1 \right)}}{2} + \frac{\log{\left(x + 1 \right)}}{2}$$
The graph
The answer [src]
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Numerical answer [src]
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The graph
Integral of x*ln((1-x)/(1+x)) dx

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