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Integral of d{x}:
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Integral of x(1-x)^5
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Identical expressions
x(one -x)^ five
x(1 minus x) to the power of 5
x(one minus x) to the power of five
x(1-x)5
x1-x5
x(1-x)⁵
x1-x^5
x(1-x)^5dx
Similar expressions
x(1+x)^5

Solving integrals/ x(1-x)^5

Integral of x(1-x)^5 dx

The solution

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  1              
  /              
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 |           5   
 |  x*(1 - x)  dx
 |               
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0

$$\int\limits_{0}^{1} x \left(1 - x\right)^{5}\, dx$$

Integral(x*(1 - x)^5, (x, 0, 1))

Detail solution

Rewrite the integrand:

$x \left(1 - x\right)^{5} = - x^{6} + 5 x^{5} - 10 x^{4} + 10 x^{3} - 5 x^{2} + x$
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- x^{6}\right)\, dx = - \int x^{6}\, dx$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{6}\, dx = \frac{x^{7}}{7}$
  So, the result is: $- \frac{x^{7}}{7}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 5 x^{5}\, dx = 5 \int x^{5}\, dx$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{5}\, dx = \frac{x^{6}}{6}$
  So, the result is: $\frac{5 x^{6}}{6}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 10 x^{4}\right)\, dx = - 10 \int x^{4}\, dx$
  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: $- 2 x^{5}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 10 x^{3}\, dx = 10 \int x^{3}\, dx$
  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: $\frac{5 x^{4}}{2}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 5 x^{2}\right)\, dx = - 5 \int x^{2}\, dx$
  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: $- \frac{5 x^{3}}{3}$
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: $- \frac{x^{7}}{7} + \frac{5 x^{6}}{6} - 2 x^{5} + \frac{5 x^{4}}{2} - \frac{5 x^{3}}{3} + \frac{x^{2}}{2}$
Now simplify:

$\frac{x^{2} \left(- 6 x^{5} + 35 x^{4} - 84 x^{3} + 105 x^{2} - 70 x + 21\right)}{42}$
Add the constant of integration:

$\frac{x^{2} \left(- 6 x^{5} + 35 x^{4} - 84 x^{3} + 105 x^{2} - 70 x + 21\right)}{42}+ \mathrm{constant}$

The answer is:

$\frac{x^{2} \left(- 6 x^{5} + 35 x^{4} - 84 x^{3} + 105 x^{2} - 70 x + 21\right)}{42}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                       
 |                      2             3    7      4      6
 |          5          x       5   5*x    x    5*x    5*x 
 | x*(1 - x)  dx = C + -- - 2*x  - ---- - -- + ---- + ----
 |                     2            3     7     2      6  
/

$$\int x \left(1 - x\right)^{5}\, dx = C - \frac{x^{7}}{7} + \frac{5 x^{6}}{6} - 2 x^{5} + \frac{5 x^{4}}{2} - \frac{5 x^{3}}{3} + \frac{x^{2}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

1/42

$$\frac{1}{42}$$

1/42

$$\frac{1}{42}$$

1/42

Numerical answer [src]

0.0238095238095238

0.0238095238095238

The graph

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

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Integral of x(1-x)^5 dx

Limits of integration:

The graph:

Piecewise:

The solution