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Solving integrals/ x*(x-1)^2

Integral of x*(x-1)^2 dx

The solution

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  1              
  /              
 |               
 |           2   
 |  x*(x - 1)  dx
 |               
/                
0

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

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

Detail solution

Rewrite the integrand:

$x \left(x - 1\right)^{2} = x^{3} - 2 x^{2} + x$
Integrate term-by-term:
1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int x^{3}\, dx = \frac{x^{4}}{4}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 2 x^{2}\right)\, dx = - 2 \int x^{2}\, dx$
  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: $- \frac{2 x^{3}}{3}$
1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int x\, dx = \frac{x^{2}}{2}$
The result is: $\frac{x^{4}}{4} - \frac{2 x^{3}}{3} + \frac{x^{2}}{2}$
Now simplify:

$\frac{x^{2} \left(3 x^{2} - 8 x + 6\right)}{12}$
Add the constant of integration:

$\frac{x^{2} \left(3 x^{2} - 8 x + 6\right)}{12}+ \mathrm{constant}$

The answer is:

$\frac{x^{2} \left(3 x^{2} - 8 x + 6\right)}{12}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                  
 |                      2      3    4
 |          2          x    2*x    x 
 | x*(x - 1)  dx = C + -- - ---- + --
 |                     2     3     4 
/

$$\int x \left(x - 1\right)^{2}\, dx = C + \frac{x^{4}}{4} - \frac{2 x^{3}}{3} + \frac{x^{2}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

1/12

$$\frac{1}{12}$$

1/12

$$\frac{1}{12}$$

1/12

Numerical answer [src]

0.0833333333333333

0.0833333333333333

The graph

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

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Integral of x*(x-1)^2 dx

Limits of integration:

The graph:

Piecewise:

The solution