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Integral of x(1/2x-1) dx

The solution

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  2             
  /             
 |              
 |    /x    \   
 |  x*|- - 1| dx
 |    \2    /   
 |              
/               
0

$$\int\limits_{0}^{2} x \left(\frac{x}{2} - 1\right)\, dx$$

Integral(x*(x/2 - 1), (x, 0, 2))

Detail solution

Rewrite the integrand:

$x \left(\frac{x}{2} - 1\right) = \frac{x^{2}}{2} - x$
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{x^{2}}{2}\, dx = \frac{\int x^{2}\, dx}{2}$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{2}\, dx = \frac{x^{3}}{3}$
  So, the result is: $\frac{x^{3}}{6}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- x\right)\, dx = - \int x\, dx$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x\, dx = \frac{x^{2}}{2}$
  So, the result is: $- \frac{x^{2}}{2}$
The result is: $\frac{x^{3}}{6} - \frac{x^{2}}{2}$
Now simplify:

$\frac{x^{2} \left(x - 3\right)}{6}$
Add the constant of integration:

$\frac{x^{2} \left(x - 3\right)}{6}+ \mathrm{constant}$

The answer is:

$\frac{x^{2} \left(x - 3\right)}{6}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                          
 |                     2    3
 |   /x    \          x    x 
 | x*|- - 1| dx = C - -- + --
 |   \2    /          2    6 
 |                           
/

$$\int x \left(\frac{x}{2} - 1\right)\, dx = C + \frac{x^{3}}{6} - \frac{x^{2}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-2/3

$$- \frac{2}{3}$$

-2/3

$$- \frac{2}{3}$$

-2/3

Numerical answer [src]

-0.666666666666667

-0.666666666666667

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

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Integral of x(1/2x-1) dx

Limits of integration:

The graph:

Piecewise:

The solution