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

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

The solution

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

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

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

Detail solution

Rewrite the integrand:

$\left(- x^{2} + x\right)^{2} = x^{4} - 2 x^{3} + x^{2}$
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:
  
  $\int \left(- 2 x^{3}\right)\, dx = - 2 \int x^{3}\, dx$
  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: $- \frac{x^{4}}{2}$
1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int x^{2}\, dx = \frac{x^{3}}{3}$
The result is: $\frac{x^{5}}{5} - \frac{x^{4}}{2} + \frac{x^{3}}{3}$
Now simplify:

$\frac{x^{3} \cdot \left(6 x^{2} - 15 x + 10\right)}{30}$
Add the constant of integration:

$\frac{x^{3} \cdot \left(6 x^{2} - 15 x + 10\right)}{30}+ \mathrm{constant}$

The answer is:

$\frac{x^{3} \cdot \left(6 x^{2} - 15 x + 10\right)}{30}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

16
--
15

$$\frac{16}{15}$$

16
--
15

$$\frac{16}{15}$$

Упростить

Numerical answer [src]

1.06666666666667

1.06666666666667

The graph

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

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

Limits of integration:

The graph:

Piecewise:

The solution