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Integral of x+2*y dx

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  2             
 x              
  /             
 |              
 |  (x + 2*y) dx
 |              
/               
x

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

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

Detail solution

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

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                    2        
 |                    x         
 | (x + 2*y) dx = C + -- + 2*x*y
 |                    2         
/

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

Simplify

The answer [src]

 4    2                 
x    x                 2
-- - -- - 2*x*y + 2*y*x 
2    2

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

 4    2                 
x    x                 2
-- - -- - 2*x*y + 2*y*x 
2    2

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

x^4/2 - x^2/2 - 2*x*y + 2*y*x^2

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