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Similar expressions
y*(x^2/2-x)

Integral of y*(x^2/2+x) dy

The solution

You have entered [src]

  1              
  /              
 |               
 |    / 2    \   
 |    |x     |   
 |  y*|-- + x| dy
 |    \2     /   
 |               
/                
0

$$\int\limits_{0}^{1} y \left(\frac{x^{2}}{2} + x\right)\, dy$$

Integral(y*(x^2/2 + x), (y, 0, 1))

Detail solution

The integral of a constant times a function is the constant times the integral of the function:

$\int y \left(\frac{x^{2}}{2} + x\right)\, dy = \left(\frac{x^{2}}{2} + x\right) \int y\, dy$
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: $\frac{y^{2} \left(\frac{x^{2}}{2} + x\right)}{2}$
Now simplify:

$\frac{x y^{2} \left(x + 2\right)}{4}$
Add the constant of integration:

$\frac{x y^{2} \left(x + 2\right)}{4}+ \mathrm{constant}$

The answer is:

$\frac{x y^{2} \left(x + 2\right)}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                       / 2    \
 |                      2 |x     |
 |   / 2    \          y *|-- + x|
 |   |x     |             \2     /
 | y*|-- + x| dy = C + -----------
 |   \2     /               2     
 |                                
/

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

Simplify

The answer [src]

     2
x   x 
- + --
2   4

$$\frac{x^{2}}{4} + \frac{x}{2}$$

     2
x   x 
- + --
2   4

$$\frac{x^{2}}{4} + \frac{x}{2}$$

x/2 + x^2/4

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

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Integral of y*(x^2/2+x) dy

Limits of integration:

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Piecewise:

The solution