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one / twelve *(x+2y)
1 divide by 12 multiply by (x plus 2y)
one divide by twelve multiply by (x plus 2y)
1/12(x+2y)
1/12x+2y
1 divide by 12*(x+2y)
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Similar expressions
1/12*(x-2y)

Integral of 1/12*(x+2y) dy

The solution

You have entered [src]

  2           
  /           
 |            
 |  x + 2*y   
 |  ------- dy
 |     12     
 |            
/             
0

$$\int\limits_{0}^{2} \frac{x + 2 y}{12}\, dy$$

Integral((x + 2*y)/12, (y, 0, 2))

Detail solution

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

$\int \frac{x + 2 y}{12}\, dy = \frac{\int \left(x + 2 y\right)\, dy}{12}$
1. Integrate term-by-term:
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int x\, dy = x y$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 2 y\, dy = 2 \int y\, dy$
    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: $y^{2}$
  The result is: $x y + y^{2}$
So, the result is: $\frac{x y}{12} + \frac{y^{2}}{12}$
Now simplify:

$\frac{y \left(x + y\right)}{12}$
Add the constant of integration:

$\frac{y \left(x + y\right)}{12}+ \mathrm{constant}$

The answer is:

$\frac{y \left(x + y\right)}{12}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                         
 |                   2      
 | x + 2*y          y    x*y
 | ------- dy = C + -- + ---
 |    12            12    12
 |                          
/

$$\int \frac{x + 2 y}{12}\, dy = C + \frac{x y}{12} + \frac{y^{2}}{12}$$

Simplify

The answer [src]

1   x
- + -
3   6

$$\frac{x}{6} + \frac{1}{3}$$

1   x
- + -
3   6

$$\frac{x}{6} + \frac{1}{3}$$

1/3 + x/6

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

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Integral of 1/12*(x+2y) dy

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The solution