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Integral of xlnx^2 dx

Limits of integration:

from to
v

The graph:

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Piecewise:

The solution

You have entered [src]
  1             
  /             
 |              
 |       2      
 |  x*log (x) dx
 |              
/               
0               
$$\int\limits_{0}^{1} x \log{\left(x \right)}^{2}\, dx$$
Integral(x*log(x)^2, (x, 0, 1))
Detail solution
  1. Let .

    Then let and substitute :

    1. Use integration by parts:

      Let and let .

      Then .

      To find :

      1. There are multiple ways to do this integral.

        Method #1

        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:

        Method #2

        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 a constant is the constant times the variable of integration:

            So, the result is:

          Now substitute back in:

      Now evaluate the sub-integral.

    2. 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.

    3. 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:

  2. Now simplify:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                              
 |                     2    2    2       2       
 |      2             x    x *log (x)   x *log(x)
 | x*log (x) dx = C + -- + ---------- - ---------
 |                    4        2            2    
/                                                
$${{x^2\,\left(2\,\left(\log x\right)^2-2\,\log x+1\right)}\over{4}}$$
The answer [src]
1/4
$${{1}\over{4}}$$
=
=
1/4
$$\frac{1}{4}$$
Numerical answer [src]
0.25
0.25

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