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Integral of x(x+2)^2 dx

The solution

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  1              
  /              
 |               
 |           2   
 |  x*(x + 2)  dx
 |               
/                
0

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

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

Detail solution

Rewrite the integrand:

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

$\frac{x^{2} \left(3 x^{2} + 16 x + 24\right)}{12}$
Add the constant of integration:

$\frac{x^{2} \left(3 x^{2} + 16 x + 24\right)}{12}+ \mathrm{constant}$

The answer is:

$\frac{x^{2} \left(3 x^{2} + 16 x + 24\right)}{12}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                    
 |                             4      3
 |          2             2   x    4*x 
 | x*(x + 2)  dx = C + 2*x  + -- + ----
 |                            4     3  
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

43
--
12

$$\frac{43}{12}$$

43
--
12

$$\frac{43}{12}$$

43/12

Numerical answer [src]

3.58333333333333

3.58333333333333

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+2)^2 dx

Limits of integration:

The graph:

Piecewise:

The solution