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1/18*x^2*(x+a)

Integral of 1/18x^2(x-a) dx

The solution

You have entered [src]

 6 + a             
   /               
  |                
  |    2           
  |   x            
  |   --*(x - a) dx
  |   18           
  |                
 /                 
 a

$$\int\limits_{a}^{a + 6} \frac{x^{2}}{18} \left(- a + x\right)\, dx$$

Integral((x^2/18)*(x - a), (x, a, 6 + a))

Detail solution

Rewrite the integrand:

$\frac{x^{2}}{18} \left(- a + x\right) = - \frac{a x^{2}}{18} + \frac{x^{3}}{18}$
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{a x^{2}}{18}\right)\, dx = - \frac{a \int x^{2}\, dx}{18}$
  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: $- \frac{a x^{3}}{54}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{x^{3}}{18}\, dx = \frac{\int x^{3}\, dx}{18}$
  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: $\frac{x^{4}}{72}$
The result is: $- \frac{a x^{3}}{54} + \frac{x^{4}}{72}$
Now simplify:

$x^{3} \left(- \frac{a}{54} + \frac{x}{72}\right)$
Add the constant of integration:

$x^{3} \left(- \frac{a}{54} + \frac{x}{72}\right)+ \mathrm{constant}$

The answer is:

$x^{3} \left(- \frac{a}{54} + \frac{x}{72}\right)+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                             
 |                              
 |  2                   4      3
 | x                   x    a*x 
 | --*(x - a) dx = C + -- - ----
 | 18                  72    54 
 |                              
/

$$\int \frac{x^{2}}{18} \left(- a + x\right)\, dx = C - \frac{a x^{3}}{54} + \frac{x^{4}}{72}$$

Simplify

The answer [src]

       4     4            3
(6 + a)     a    a*(6 + a) 
-------- + --- - ----------
   72      216       54

$$\frac{a^{4}}{216} - \frac{a \left(a + 6\right)^{3}}{54} + \frac{\left(a + 6\right)^{4}}{72}$$

       4     4            3
(6 + a)     a    a*(6 + a) 
-------- + --- - ----------
   72      216       54

$$\frac{a^{4}}{216} - \frac{a \left(a + 6\right)^{3}}{54} + \frac{\left(a + 6\right)^{4}}{72}$$

(6 + a)^4/72 + a^4/216 - a*(6 + a)^3/54

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

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Integral of 1/18x^2(x-a) dx

Limits of integration:

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Piecewise:

The solution

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Integral of 1/18*x^2*(x-a) dx

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Integral of 1/18x^2(x-a) dx