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Integral of xy(x+y) dy

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 2 - 2*x              
    /                 
   |                  
   |    x*y*(x + y) dy
   |                  
  /                   
  0

$$\int\limits_{0}^{2 - 2 x} x y \left(x + y\right)\, dy$$

Integral((x*y)*(x + y), (y, 0, 2 - 2*x))

Detail solution

Rewrite the integrand:

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

$\frac{x y^{2} \left(3 x + 2 y\right)}{6}$
Add the constant of integration:

$\frac{x y^{2} \left(3 x + 2 y\right)}{6}+ \mathrm{constant}$

The answer is:

$\frac{x y^{2} \left(3 x + 2 y\right)}{6}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                      2  2      3
 |                      x *y    x*y 
 | x*y*(x + y) dy = C + ----- + ----
 |                        2      3  
/

$$\int x y \left(x + y\right)\, dy = C + \frac{x^{2} y^{2}}{2} + \frac{x y^{3}}{3}$$

Simplify

The answer [src]

 2          2              3
x *(2 - 2*x)    x*(2 - 2*x) 
------------- + ------------
      2              3

$$\frac{x^{2} \left(2 - 2 x\right)^{2}}{2} + \frac{x \left(2 - 2 x\right)^{3}}{3}$$

 2          2              3
x *(2 - 2*x)    x*(2 - 2*x) 
------------- + ------------
      2              3

$$\frac{x^{2} \left(2 - 2 x\right)^{2}}{2} + \frac{x \left(2 - 2 x\right)^{3}}{3}$$

x^2*(2 - 2*x)^2/2 + x*(2 - 2*x)^3/3

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

Integral of x*y*(x+y) dy