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Similar expressions
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Solving integrals/ x^3*(2-x)^2

Integral of x^3*(2-x)^2 dx

The solution

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  2               
  /               
 |                
 |   3        2   
 |  x *(2 - x)  dx
 |                
/                 
0

$$\int\limits_{0}^{2} x^{3} \left(2 - x\right)^{2}\, dx$$

Integral(x^3*(2 - x)^2, (x, 0, 2))

Detail solution

Rewrite the integrand:

$x^{3} \left(2 - x\right)^{2} = x^{5} - 4 x^{4} + 4 x^{3}$
Integrate term-by-term:
1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int x^{5}\, dx = \frac{x^{6}}{6}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 4 x^{4}\right)\, dx = - 4 \int x^{4}\, dx$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{4}\, dx = \frac{x^{5}}{5}$
  So, the result is: $- \frac{4 x^{5}}{5}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 4 x^{3}\, dx = 4 \int x^{3}\, dx$
  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: $x^{4}$
The result is: $\frac{x^{6}}{6} - \frac{4 x^{5}}{5} + x^{4}$
Now simplify:

$\frac{x^{4} \cdot \left(5 x^{2} - 24 x + 30\right)}{30}$
Add the constant of integration:

$\frac{x^{4} \cdot \left(5 x^{2} - 24 x + 30\right)}{30}+ \mathrm{constant}$

The answer is:

$\frac{x^{4} \cdot \left(5 x^{2} - 24 x + 30\right)}{30}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                   
 |                              5    6
 |  3        2           4   4*x    x 
 | x *(2 - x)  dx = C + x  - ---- + --
 |                            5     6 
/

$${{5\,x^6-24\,x^5+30\,x^4}\over{30}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

16
--
15

$${{16}\over{15}}$$

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^3*(2-x)^2 dx

Limits of integration:

The graph:

Piecewise:

The solution