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x*(x/2-1/2)

Solving integrals/ x*(x/2+1/2)

Integral of x*(x/2+1/2) dx

The solution

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

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

Integral(x*(x/2 + 1/2), (x, -1, 1))

Detail solution

Rewrite the integrand:

$x \left(\frac{x}{2} + \frac{1}{2}\right) = \frac{x^{2}}{2} + \frac{x}{2}$
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{x^{2}}{2}\, dx = \frac{\int x^{2}\, dx}{2}$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{2}\, dx = \frac{x^{3}}{3}$
  So, the result is: $\frac{x^{3}}{6}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{x}{2}\, dx = \frac{\int x\, dx}{2}$
  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: $\frac{x^{2}}{4}$
The result is: $\frac{x^{3}}{6} + \frac{x^{2}}{4}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                          
 |                     2    3
 |   /x   1\          x    x 
 | x*|- + -| dx = C + -- + --
 |   \2   2/          4    6 
 |                           
/

$$\int x \left(\frac{x}{2} + \frac{1}{2}\right)\, dx = C + \frac{x^{3}}{6} + \frac{x^{2}}{4}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

1/3

$$\frac{1}{3}$$

1/3

$$\frac{1}{3}$$

Упростить

Numerical answer [src]

0.333333333333333

0.333333333333333

The graph

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

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Integral of x*(x/2+1/2) dx

Limits of integration:

The graph:

Piecewise:

The solution