Mister Exam

Integral of xln(x-1) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

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

    2. Integrate term-by-term:

      1. The integral of is when :

      1. The integral of a constant is the constant times the variable of integration:

      1. Let .

        Then let and substitute :

        1. The integral of is .

        Now substitute back in:

      The result is:

    So, the result is:

  3. Now simplify:

  4. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                         2    2           
 |                       x   log(-1 + x)   x    x *log(x - 1)
 | x*log(x - 1) dx = C - - - ----------- - -- + -------------
 |                       2        2        4          2      
/                                                            
$$\int x \log{\left(x - 1 \right)}\, dx = C + \frac{x^{2} \log{\left(x - 1 \right)}}{2} - \frac{x^{2}}{4} - \frac{x}{2} - \frac{\log{\left(x - 1 \right)}}{2}$$
The graph
The answer [src]
  3   pi*I
- - + ----
  4    2  
$$- \frac{3}{4} + \frac{i \pi}{2}$$
=
=
  3   pi*I
- - + ----
  4    2  
$$- \frac{3}{4} + \frac{i \pi}{2}$$
-3/4 + pi*i/2
Numerical answer [src]
(-0.75 + 1.5707963267949j)
(-0.75 + 1.5707963267949j)
The graph
Integral of xln(x-1) dx

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