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Integral of x^2logxdx dx

Limits of integration:

from to
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Piecewise:

The solution

You have entered [src]
  1               
  /               
 |                
 |   2            
 |  x *log(x)*1 dx
 |                
/                 
0                 
$$\int\limits_{0}^{1} x^{2} \log{\left(x \right)} 1\, dx$$
Integral(x^2*log(x)*1, (x, 0, 1))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    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. 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:

    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. The integral of is when :

      So, the result is:

  2. Now simplify:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                   
 |                       3    3       
 |  2                   x    x *log(x)
 | x *log(x)*1 dx = C - -- + ---------
 |                      9        3    
/                                     
$${{x^3\,\log x}\over{3}}-{{x^3}\over{9}}$$
The answer [src]
-1/9
$$-{{1}\over{9}}$$
=
=
-1/9
$$- \frac{1}{9}$$
Numerical answer [src]
-0.111111111111111
-0.111111111111111

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