Mister Exam

Integral of x^2ln2xdx dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

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

    Method #1

    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. The integral of is when :

        So, the result is:

      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. The integral of is when :

      So, the result is:

    Method #3

    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. The integral of is when :

        So, the result is:

      The result is:

  2. Now simplify:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                               
 |                       3    3           3       
 |  2                   x    x *log(2)   x *log(x)
 | x *log(2*x) dx = C - -- + --------- + ---------
 |                      9        3           3    
/                                                 
$$\int x^{2} \log{\left(2 x \right)}\, dx = C + \frac{x^{3} \log{\left(x \right)}}{3} - \frac{x^{3}}{9} + \frac{x^{3} \log{\left(2 \right)}}{3}$$
The graph
The answer [src]
              3    3         
1   log(2)   e    e *log(2*E)
- - ------ - -- + -----------
9     3      9         3     
$$- \frac{e^{3}}{9} - \frac{\log{\left(2 \right)}}{3} + \frac{1}{9} + \frac{e^{3} \log{\left(2 e \right)}}{3}$$
=
=
              3    3         
1   log(2)   e    e *log(2*E)
- - ------ - -- + -----------
9     3      9         3     
$$- \frac{e^{3}}{9} - \frac{\log{\left(2 \right)}}{3} + \frac{1}{9} + \frac{e^{3} \log{\left(2 e \right)}}{3}$$
1/9 - log(2)/3 - exp(3)/9 + exp(3)*log(2*E)/3
Numerical answer [src]
8.98425912996846
8.98425912996846

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