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x^3*log(x)

Integral of x^3*log(x) dx

Limits of integration:

from to
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The graph:

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Piecewise:

The solution

You have entered [src]
  1             
  /             
 |              
 |   3          
 |  x *log(x) dx
 |              
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0               
$$\int\limits_{0}^{1} x^{3} \log{\left(x \right)}\, dx$$
Integral(x^3*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. 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:

    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]
  /                                 
 |                     4    4       
 |  3                 x    x *log(x)
 | x *log(x) dx = C - -- + ---------
 |                    16       4    
/                                   
$$\int x^{3} \log{\left(x \right)}\, dx = C + \frac{x^{4} \log{\left(x \right)}}{4} - \frac{x^{4}}{16}$$
The graph
The answer [src]
-1/16
$$- \frac{1}{16}$$
=
=
-1/16
$$- \frac{1}{16}$$
-1/16
Numerical answer [src]
-0.0625
-0.0625
The graph
Integral of x^3*log(x) dx

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