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Identical expressions
x(x- five)^ twelve
x(x minus 5) to the power of 12
x(x minus five) to the power of twelve
x(x-5)12
xx-512
xx-5^12
x(x-5)^12dx
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x(x+5)^12

Solving integrals/ x(x-5)^12

Integral of x(x-5)^12 dx

The solution

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  6               
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 |           12   
 |  x*(x - 5)   dx
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5

$$\int\limits_{5}^{6} x \left(x - 5\right)^{12}\, dx$$

Integral(x*(x - 5)^12, (x, 5, 6))

Detail solution

Rewrite the integrand:

$x \left(x - 5\right)^{12} = x^{13} - 60 x^{12} + 1650 x^{11} - 27500 x^{10} + 309375 x^{9} - 2475000 x^{8} + 14437500 x^{7} - 61875000 x^{6} + 193359375 x^{5} - 429687500 x^{4} + 644531250 x^{3} - 585937500 x^{2} + 244140625 x$
Integrate term-by-term:
1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int x^{13}\, dx = \frac{x^{14}}{14}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 60 x^{12}\right)\, dx = - 60 \int x^{12}\, dx$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{12}\, dx = \frac{x^{13}}{13}$
  So, the result is: $- \frac{60 x^{13}}{13}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 1650 x^{11}\, dx = 1650 \int x^{11}\, dx$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{11}\, dx = \frac{x^{12}}{12}$
  So, the result is: $\frac{275 x^{12}}{2}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 27500 x^{10}\right)\, dx = - 27500 \int x^{10}\, dx$
  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: $- 2500 x^{11}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 309375 x^{9}\, dx = 309375 \int x^{9}\, dx$
  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: $\frac{61875 x^{10}}{2}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 2475000 x^{8}\right)\, dx = - 2475000 \int x^{8}\, dx$
  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: $- 275000 x^{9}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 14437500 x^{7}\, dx = 14437500 \int x^{7}\, dx$
  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: $\frac{3609375 x^{8}}{2}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 61875000 x^{6}\right)\, dx = - 61875000 \int x^{6}\, dx$
  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: $- \frac{61875000 x^{7}}{7}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 193359375 x^{5}\, dx = 193359375 \int x^{5}\, dx$
  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: $\frac{64453125 x^{6}}{2}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 429687500 x^{4}\right)\, dx = - 429687500 \int x^{4}\, dx$
  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: $- 85937500 x^{5}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 644531250 x^{3}\, dx = 644531250 \int x^{3}\, dx$
  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: $\frac{322265625 x^{4}}{2}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 585937500 x^{2}\right)\, dx = - 585937500 \int x^{2}\, dx$
  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: $- 195312500 x^{3}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 244140625 x\, dx = 244140625 \int x\, dx$
  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: $\frac{244140625 x^{2}}{2}$
The result is: $\frac{x^{14}}{14} - \frac{60 x^{13}}{13} + \frac{275 x^{12}}{2} - 2500 x^{11} + \frac{61875 x^{10}}{2} - 275000 x^{9} + \frac{3609375 x^{8}}{2} - \frac{61875000 x^{7}}{7} + \frac{64453125 x^{6}}{2} - 85937500 x^{5} + \frac{322265625 x^{4}}{2} - 195312500 x^{3} + \frac{244140625 x^{2}}{2}$
Now simplify:

$\frac{x^{2} \left(13 x^{12} - 840 x^{11} + 25025 x^{10} - 455000 x^{9} + 5630625 x^{8} - 50050000 x^{7} + 328453125 x^{6} - 1608750000 x^{5} + 5865234375 x^{4} - 15640625000 x^{3} + 29326171875 x^{2} - 35546875000 x + 22216796875\right)}{182}$
Add the constant of integration:

$\frac{x^{2} \left(13 x^{12} - 840 x^{11} + 25025 x^{10} - 455000 x^{9} + 5630625 x^{8} - 50050000 x^{7} + 328453125 x^{6} - 1608750000 x^{5} + 5865234375 x^{4} - 15640625000 x^{3} + 29326171875 x^{2} - 35546875000 x + 22216796875\right)}{182}+ \mathrm{constant}$

The answer is:

$\frac{x^{2} \left(13 x^{12} - 840 x^{11} + 25025 x^{10} - 455000 x^{9} + 5630625 x^{8} - 50050000 x^{7} + 328453125 x^{6} - 1608750000 x^{5} + 5865234375 x^{4} - 15640625000 x^{3} + 29326171875 x^{2} - 35546875000 x + 22216796875\right)}{182}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                                                                                                                                  
 |                                                                                    7       13    14        12          10            8             6              2              4
 |          12                     3             5           9         11   61875000*x    60*x     x     275*x     61875*x     3609375*x    64453125*x    244140625*x    322265625*x 
 | x*(x - 5)   dx = C - 195312500*x  - 85937500*x  - 275000*x  - 2500*x   - ----------- - ------ + --- + ------- + --------- + ---------- + ----------- + ------------ + ------------
 |                                                                               7          13      14      2          2           2             2             2              2      
/

$$\int x \left(x - 5\right)^{12}\, dx = C + \frac{x^{14}}{14} - \frac{60 x^{13}}{13} + \frac{275 x^{12}}{2} - 2500 x^{11} + \frac{61875 x^{10}}{2} - 275000 x^{9} + \frac{3609375 x^{8}}{2} - \frac{61875000 x^{7}}{7} + \frac{64453125 x^{6}}{2} - 85937500 x^{5} + \frac{322265625 x^{4}}{2} - 195312500 x^{3} + \frac{244140625 x^{2}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

 83
---
182

$$\frac{83}{182}$$

 83
---
182

$$\frac{83}{182}$$

83/182

Numerical answer [src]

0.456043956043956

0.456043956043956

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Integral of x(x-5)^12 dx

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