Integral of (x²-2x)² dx

The solution

You have entered [src]

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

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

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

Detail solution

Rewrite the integrand:

$\left(x^{2} - 2 x\right)^{2} = x^{4} - 4 x^{3} + 4 x^{2}$
Integrate term-by-term:
1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int x^{4}\, dx = \frac{x^{5}}{5}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 4 x^{3}\right)\, dx = - 4 \int x^{3}\, dx$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{3}\, dx = \frac{x^{4}}{4}$
  So, the result is: $- x^{4}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 4 x^{2}\, dx = 4 \int x^{2}\, dx$
  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: $\frac{4 x^{3}}{3}$
The result is: $\frac{x^{5}}{5} - x^{4} + \frac{4 x^{3}}{3}$
Now simplify:

$x^{3} \left(\frac{x^{2}}{5} - x + \frac{4}{3}\right)$
Add the constant of integration:

$x^{3} \left(\frac{x^{2}}{5} - x + \frac{4}{3}\right)+ \mathrm{constant}$

The answer is:

$x^{3} \left(\frac{x^{2}}{5} - x + \frac{4}{3}\right)+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                   
 |                                    
 |           2                5      3
 | / 2      \            4   x    4*x 
 | \x  - 2*x/  dx = C - x  + -- + ----
 |                           5     3  
/

$$\int \left(x^{2} - 2 x\right)^{2}\, dx = C + \frac{x^{5}}{5} - x^{4} + \frac{4 x^{3}}{3}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

8/15

$$\frac{8}{15}$$

8/15

$$\frac{8}{15}$$

8/15

Numerical answer [src]

0.533333333333333

0.533333333333333

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of (x²-2x)² dx

Limits of integration:

The graph:

Piecewise:

The solution