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Integral of y+y^2 dy

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  2            
  /            
 |             
 |  /     2\   
 |  \y + y / dy
 |             
/              
x

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

Integral(y + y^2, (y, x, 2))

Detail solution

Integrate term-by-term:
1. The integral of $y^{n}$ is $\frac{y^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int y^{2}\, dy = \frac{y^{3}}{3}$
1. The integral of $y^{n}$ is $\frac{y^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int y\, dy = \frac{y^{2}}{2}$
The result is: $\frac{y^{3}}{3} + \frac{y^{2}}{2}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                         
 |                    2    3
 | /     2\          y    y 
 | \y + y / dy = C + -- + --
 |                   2    3 
/

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

Simplify

The answer [src]

      2    3
14   x    x 
-- - -- - --
3    2    3

$$- \frac{x^{3}}{3} - \frac{x^{2}}{2} + \frac{14}{3}$$

      2    3
14   x    x 
-- - -- - --
3    2    3

$$- \frac{x^{3}}{3} - \frac{x^{2}}{2} + \frac{14}{3}$$

14/3 - x^2/2 - x^3/3

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