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Integral of d{x}:
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Integral of x(1-x)^10
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Integral of (x+1)^4/(x^2+2x+2)^3
Identical expressions
x(one -x)^ ten
x(1 minus x) to the power of 10
x(one minus x) to the power of ten
x(1-x)10
x1-x10
x1-x^10
x(1-x)^10dx
Similar expressions
x(1+x)^10

Solving integrals/ x(1-x)^10

Integral of x(1-x)^10 dx

The solution

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$$\int\limits_{0}^{1} x \left(1 - x\right)^{10}\, dx$$

Integral(x*(1 - x)^10, (x, 0, 1))

Detail solution

Rewrite the integrand:

$x \left(1 - x\right)^{10} = x^{11} - 10 x^{10} + 45 x^{9} - 120 x^{8} + 210 x^{7} - 252 x^{6} + 210 x^{5} - 120 x^{4} + 45 x^{3} - 10 x^{2} + x$
Integrate term-by-term:
1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int x^{11}\, dx = \frac{x^{12}}{12}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 10 x^{10}\right)\, dx = - 10 \int x^{10}\, dx$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{10}\, dx = \frac{x^{11}}{11}$
  So, the result is: $- \frac{10 x^{11}}{11}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 45 x^{9}\, dx = 45 \int x^{9}\, dx$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{9}\, dx = \frac{x^{10}}{10}$
  So, the result is: $\frac{9 x^{10}}{2}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 120 x^{8}\right)\, dx = - 120 \int x^{8}\, dx$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{8}\, dx = \frac{x^{9}}{9}$
  So, the result is: $- \frac{40 x^{9}}{3}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 210 x^{7}\, dx = 210 \int x^{7}\, dx$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{7}\, dx = \frac{x^{8}}{8}$
  So, the result is: $\frac{105 x^{8}}{4}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 252 x^{6}\right)\, dx = - 252 \int x^{6}\, dx$
  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: $- 36 x^{7}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 210 x^{5}\, dx = 210 \int x^{5}\, dx$
  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: $35 x^{6}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 120 x^{4}\right)\, dx = - 120 \int x^{4}\, dx$
  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: $- 24 x^{5}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 45 x^{3}\, dx = 45 \int x^{3}\, dx$
  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: $\frac{45 x^{4}}{4}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 10 x^{2}\right)\, dx = - 10 \int x^{2}\, dx$
  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: $- \frac{10 x^{3}}{3}$
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: $\frac{x^{12}}{12} - \frac{10 x^{11}}{11} + \frac{9 x^{10}}{2} - \frac{40 x^{9}}{3} + \frac{105 x^{8}}{4} - 36 x^{7} + 35 x^{6} - 24 x^{5} + \frac{45 x^{4}}{4} - \frac{10 x^{3}}{3} + \frac{x^{2}}{2}$
Now simplify:

$\frac{x^{2} \left(11 x^{10} - 120 x^{9} + 594 x^{8} - 1760 x^{7} + 3465 x^{6} - 4752 x^{5} + 4620 x^{4} - 3168 x^{3} + 1485 x^{2} - 440 x + 66\right)}{132}$
Add the constant of integration:

$\frac{x^{2} \left(11 x^{10} - 120 x^{9} + 594 x^{8} - 1760 x^{7} + 3465 x^{6} - 4752 x^{5} + 4620 x^{4} - 3168 x^{3} + 1485 x^{2} - 440 x + 66\right)}{132}+ \mathrm{constant}$

The answer is:

$\frac{x^{2} \left(11 x^{10} - 120 x^{9} + 594 x^{8} - 1760 x^{7} + 3465 x^{6} - 4752 x^{5} + 4620 x^{4} - 3168 x^{3} + 1485 x^{2} - 440 x + 66\right)}{132}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                                                       
 |                       2                               9       3       11    12      10       4        8
 |          10          x        7       5       6   40*x    10*x    10*x     x     9*x     45*x    105*x 
 | x*(1 - x)   dx = C + -- - 36*x  - 24*x  + 35*x  - ----- - ----- - ------ + --- + ----- + ----- + ------
 |                      2                              3       3       11      12     2       4       4   
/

$$\int x \left(1 - x\right)^{10}\, dx = C + \frac{x^{12}}{12} - \frac{10 x^{11}}{11} + \frac{9 x^{10}}{2} - \frac{40 x^{9}}{3} + \frac{105 x^{8}}{4} - 36 x^{7} + 35 x^{6} - 24 x^{5} + \frac{45 x^{4}}{4} - \frac{10 x^{3}}{3} + \frac{x^{2}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

1/132

$$\frac{1}{132}$$

1/132

$$\frac{1}{132}$$

1/132

Numerical answer [src]

0.00757575757575758

0.00757575757575758

The graph

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

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Integral of x(1-x)^10 dx

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The solution