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

The solution

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

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

Detail solution

Rewrite the integrand:

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

$x^{2} \left(\frac{x^{2}}{4} - x + 1\right)$
Add the constant of integration:

$x^{2} \left(\frac{x^{2}}{4} - x + 1\right)+ \mathrm{constant}$

The answer is:

$x^{2} \left(\frac{x^{2}}{4} - x + 1\right)+ \mathrm{constant}$

The answer (Indefinite) [src]

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

1/4

$$\frac{1}{4}$$

1/4

$$\frac{1}{4}$$

1/4

Numerical answer [src]

0.25

0.25

The graph

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

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Identical expressions

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

Limits of integration:

The graph:

Piecewise:

The solution