Mister Exam

Integral of 2xlnxdx dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1                
  /                
 |                 
 |  2*x*log(x)*1 dx
 |                 
/                  
0                  
$$\int\limits_{0}^{1} 2 x \log{\left(x \right)} 1\, dx$$
Detail solution
  1. The integral of a constant times a function is the constant times the integral of the function:

    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:

    So, the result is:

  2. Now simplify:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                       2            
 |                       x     2       
 | 2*x*log(x)*1 dx = C - -- + x *log(x)
 |                       2             
/                                      
$$2\,\left({{x^2\,\log x}\over{2}}-{{x^2}\over{4}}\right)$$
The answer [src]
-1/2
$$-{{1}\over{2}}$$
=
=
-1/2
$$- \frac{1}{2}$$
Numerical answer [src]
-0.5
-0.5

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