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Solving integrals/ y*(x^2/2+x)

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

Limits of integration:

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The solution

You have entered [src] 
  1              
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0
∫01y(x22+x) dy\int\limits_{0}^{1} y \left(\frac{x^{2}}{2} + x\right)\, dy 
Integral(y*(x^2/2 + x), (y, 0, 1))
Detail solution

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

    ∫y(x22+x) dy=(x22+x)∫y dy\int y \left(\frac{x^{2}}{2} + x\right)\, dy = \left(\frac{x^{2}}{2} + x\right) \int y\, dy

    1. The integral of yny^{n} is yn+1n+1\frac{y^{n + 1}}{n + 1} when n≠−1n \neq -1:

      ∫y dy=y22\int y\, dy = \frac{y^{2}}{2}

    So, the result is: y2(x22+x)2\frac{y^{2} \left(\frac{x^{2}}{2} + x\right)}{2}

  2. Now simplify:

    xy2(x+2)4\frac{x y^{2} \left(x + 2\right)}{4}

  3. Add the constant of integration:

    xy2(x+2)4+constant\frac{x y^{2} \left(x + 2\right)}{4}+ \mathrm{constant}

The answer is:

xy2(x+2)4+constant\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     
 |                                
/
∫y(x22+x) dy=C+y2(x22+x)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
x24+x2\frac{x^{2}}{4} + \frac{x}{2} 
=
= 
     2
x   x 
- + --
2   4
x24+x2\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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