Integral of (x-x^2) dx

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

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

Detail solution

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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$${{x^2}\over{2}}-{{x^3}\over{3}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

1/6

$${{1}\over{6}}$$

1/6

$$\frac{1}{6}$$

Упростить

Numerical answer [src]

0.166666666666667

0.166666666666667

The graph

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