Integral of 2xy+y dy

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

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

Detail solution

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

$y^{2} \left(x + \frac{1}{2}\right)$
Add the constant of integration:

$y^{2} \left(x + \frac{1}{2}\right)+ \mathrm{constant}$

The answer is:

$y^{2} \left(x + \frac{1}{2}\right)+ \mathrm{constant}$

The answer (Indefinite) [src]

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

Simplify

The answer [src]

2 + 4*x

$$4 x + 2$$

2 + 4*x

$$4 x + 2$$

2 + 4*x

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