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Integral of (x^2-2x)lnx dx

Limits of integration:

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Piecewise:

The solution

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$$\int\limits_{0}^{1} \left(x^{2} - 2 x\right) \log{\left(x \right)}\, dx$$
Integral((x^2 - 2*x)*log(x), (x, 0, 1))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Let .

      Then let and substitute :

      1. Integrate term-by-term:

        1. Use integration by parts:

          Let and let .

          Then .

          To find :

          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. The integral of the exponential function is itself.

              So, the result is:

            Now substitute back in:

          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. The integral of the exponential function is itself.

              So, the result 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. Use integration by parts:

            Let and let .

            Then .

            To find :

            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. The integral of the exponential function is itself.

                So, the result is:

              Now substitute back in:

            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. The integral of the exponential function is itself.

                So, the result is:

              Now substitute back in:

            So, the result is:

          So, the result is:

        The result is:

      Now substitute back in:

    Method #2

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      1. Let .

        Then let and substitute :

        1. Use integration by parts:

          Let and let .

          Then .

          To find :

          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. The integral of the exponential function is itself.

              So, the result is:

            Now substitute back in:

          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. The integral of the exponential function is itself.

              So, the result is:

            Now substitute back in:

          So, the result is:

        Now substitute back in:

      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. Use integration by parts:

            Let and let .

            Then .

            To find :

            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. The integral of the exponential function is itself.

                So, the result is:

              Now substitute back in:

            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. The integral of the exponential function is itself.

                So, the result is:

              Now substitute back in:

            So, the result is:

          Now substitute back in:

        So, the result is:

      The result is:

    Method #3

    1. Use integration by parts:

      Let and let .

      Then .

      To find :

      1. Rewrite the integrand:

      2. Integrate term-by-term:

        1. The integral of is when :

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

          1. The integral of is when :

          So, the result is:

        The result is:

      Now evaluate the sub-integral.

    2. Rewrite the integrand:

    3. Integrate term-by-term:

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

        1. The integral of is when :

        So, the result is:

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

        1. The integral of is when :

        So, the result is:

      The result is:

    Method #4

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      1. Let .

        Then let and substitute :

        1. Use integration by parts:

          Let and let .

          Then .

          To find :

          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. The integral of the exponential function is itself.

              So, the result is:

            Now substitute back in:

          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. The integral of the exponential function is itself.

              So, the result is:

            Now substitute back in:

          So, the result is:

        Now substitute back in:

      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. Use integration by parts:

            Let and let .

            Then .

            To find :

            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. The integral of the exponential function is itself.

                So, the result is:

              Now substitute back in:

            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. The integral of the exponential function is itself.

                So, the result is:

              Now substitute back in:

            So, the result is:

          Now substitute back in:

        So, the result is:

      The result is:

  2. Now simplify:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                          
 |                             2    3                3       
 | / 2      \                 x    x     2          x *log(x)
 | \x  - 2*x/*log(x) dx = C + -- - -- - x *log(x) + ---------
 |                            2    9                    3    
/                                                            
$$\int \left(x^{2} - 2 x\right) \log{\left(x \right)}\, dx = C + \frac{x^{3} \log{\left(x \right)}}{3} - \frac{x^{3}}{9} - x^{2} \log{\left(x \right)} + \frac{x^{2}}{2}$$
The graph
The answer [src]
7/18
$$\frac{7}{18}$$
=
=
7/18
$$\frac{7}{18}$$
7/18
Numerical answer [src]
0.388888888888889
0.388888888888889

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