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Solving integrals/ x^3(1-x)^2

Integral of x^3(1-x)^2 dx

The solution

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  1               
  /               
 |                
 |   3        2   
 |  x *(1 - x)  dx
 |                
/                 
0

$$\int\limits_{0}^{1} x^{3} \left(1 - x\right)^{2}\, dx$$

Integral(x^3*(1 - x)^2, (x, 0, 1))

Detail solution

Rewrite the integrand:

$x^{3} \left(1 - x\right)^{2} = x^{5} - 2 x^{4} + x^{3}$
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:
  
  $\int \left(- 2 x^{4}\right)\, dx = - 2 \int x^{4}\, dx$
  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: $- \frac{2 x^{5}}{5}$
1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int x^{3}\, dx = \frac{x^{4}}{4}$
The result is: $\frac{x^{6}}{6} - \frac{2 x^{5}}{5} + \frac{x^{4}}{4}$
Now simplify:

$\frac{x^{4} \cdot \left(10 x^{2} - 24 x + 15\right)}{60}$
Add the constant of integration:

$\frac{x^{4} \cdot \left(10 x^{2} - 24 x + 15\right)}{60}+ \mathrm{constant}$

The answer is:

$\frac{x^{4} \cdot \left(10 x^{2} - 24 x + 15\right)}{60}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                   
 |                         5    4    6
 |  3        2          2*x    x    x 
 | x *(1 - x)  dx = C - ---- + -- + --
 |                       5     4    6 
/

$$\int x^{3} \left(1 - x\right)^{2}\, dx = C + \frac{x^{6}}{6} - \frac{2 x^{5}}{5} + \frac{x^{4}}{4}$$

Simplify

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Plot the graph f(x)

The answer [src]

1/60

$$\frac{1}{60}$$

1/60

$$\frac{1}{60}$$

Упростить

Numerical answer [src]

0.0166666666666667

0.0166666666666667

The graph

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

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Integral of x^3(1-x)^2 dx

Limits of integration:

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The solution