Integral of xy(x+y) dx

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

Integral((x*y)*(x + y), (x, 0, 1))

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:
  
  $\int x^{2} y\, dx = y \int x^{2}\, dx$
  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: $\frac{x^{3} y}{3}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int x y^{2}\, dx = y^{2} \int x\, dx$
  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: $\frac{x^{2} y^{2}}{2}$
The result is: $\frac{x^{3} y}{3} + \frac{x^{2} y^{2}}{2}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The answer [src]

 2    
y    y
-- + -
2    3

$$\frac{y^{2}}{2} + \frac{y}{3}$$

 2    
y    y
-- + -
2    3

$$\frac{y^{2}}{2} + \frac{y}{3}$$

y^2/2 + y/3

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