Mister Exam

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Identical expressions
(x^ two - two xy)+(2xy+y^2)
(x squared minus 2xy) plus (2xy plus y squared )
(x to the power of two minus two xy) plus (2xy plus y squared )
(x2-2xy)+(2xy+y2)
x2-2xy+2xy+y2
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(x to the power of 2-2xy)+(2xy+y to the power of 2)
x^2-2xy+2xy+y^2
(x^2-2xy)+(2xy+y^2)dx
Similar expressions
(x^2+2xy)+(2xy+y^2)
(x^2-2xy)+(2xy-y^2)
(x^2-2xy)-(2xy+y^2)

Integral of (x^2-2xy)+(2xy+y^2) dx

The solution

You have entered [src]

  1                             
  /                             
 |                              
 |  / 2                    2\   
 |  \x  - 2*x*y + 2*x*y + y / dx
 |                              
/                               
0

$$\int\limits_{0}^{1} \left(\left(x^{2} - 2 x y\right) + \left(2 x y + y^{2}\right)\right)\, dx$$

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

Detail solution

Integrate term-by-term:
1. Integrate term-by-term:
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{2}\, dx = \frac{x^{3}}{3}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 2 x y\right)\, dx = - y \int 2 x\, dx$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 2 x\, dx = 2 \int x\, dx$
      1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
        
        $\int x\, dx = \frac{x^{2}}{2}$
      So, the result is: $x^{2}$
    So, the result is: $- x^{2} y$
  The result is: $\frac{x^{3}}{3} - x^{2} y$
1. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 2 x y\, dx = y \int 2 x\, dx$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 2 x\, dx = 2 \int x\, dx$
      1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
        
        $\int x\, dx = \frac{x^{2}}{2}$
      So, the result is: $x^{2}$
    So, the result is: $x^{2} y$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int y^{2}\, dx = x y^{2}$
  The result is: $x^{2} y + x y^{2}$
The result is: $\frac{x^{3}}{3} + x y^{2}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The answer [src]

1    2
- + y 
3

$$y^{2} + \frac{1}{3}$$

1    2
- + y 
3

$$y^{2} + \frac{1}{3}$$

1/3 + y^2

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

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Integral of (x^2-2xy)+(2xy+y^2) dx

Limits of integration:

The graph:

Piecewise:

The solution