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Identical expressions
xy^ two +(x^ two)/ eight
xy squared plus (x squared ) divide by 8
xy to the power of two plus (x to the power of two) divide by eight
xy2+(x2)/8
xy2+x2/8
xy²+(x²)/8
xy to the power of 2+(x to the power of 2)/8
xy^2+x^2/8
xy^2+(x^2) divide by 8
xy^2+(x^2)/8dx
Similar expressions
xy^2-(x^2)/8

Integral of xy^2+(x^2)/8 dy

The solution

You have entered [src]

 1/2              
  /               
 |                
 |  /        2\   
 |  |   2   x |   
 |  |x*y  + --| dy
 |  \       8 /   
 |                
/                 
0

$$\int\limits_{0}^{\frac{1}{2}} \left(\frac{x^{2}}{8} + x y^{2}\right)\, dy$$

Integral(x*y^2 + x^2/8, (y, 0, 1/2))

Detail solution

Integrate term-by-term:
1. The integral of a constant is the constant times the variable of integration:
  
  $\int \frac{x^{2}}{8}\, dy = \frac{x^{2} y}{8}$
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: $\frac{x^{2} y}{8} + \frac{x y^{3}}{3}$
Now simplify:

$\frac{x y \left(3 x + 8 y^{2}\right)}{24}$
Add the constant of integration:

$\frac{x y \left(3 x + 8 y^{2}\right)}{24}+ \mathrm{constant}$

The answer is:

$\frac{x y \left(3 x + 8 y^{2}\right)}{24}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                
 |                                 
 | /        2\             3      2
 | |   2   x |          x*y    y*x 
 | |x*y  + --| dy = C + ---- + ----
 | \       8 /           3      8  
 |                                 
/

$$\int \left(\frac{x^{2}}{8} + x y^{2}\right)\, dy = C + \frac{x^{2} y}{8} + \frac{x y^{3}}{3}$$

Simplify

The answer [src]

 2     
x    x 
-- + --
16   24

$$\frac{x^{2}}{16} + \frac{x}{24}$$

 2     
x    x 
-- + --
16   24

$$\frac{x^{2}}{16} + \frac{x}{24}$$

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

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Integral of xy^2+(x^2)/8 dy

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The solution