Mister Exam

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Integral of (x^2+y)+(2x-y) dy

The solution

You have entered [src]

  1                      
  /                      
 |                       
 |  / 2              \   
 |  \x  + y + 2*x - y/ dy
 |                       
/                        
a*b

$$\int\limits_{a b}^{1} \left(\left(2 x - y\right) + \left(x^{2} + y\right)\right)\, dy$$

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

Detail solution

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The answer [src]

 2             / 2      \
x  + 2*x - a*b*\x  + 2*x/

$$- a b \left(x^{2} + 2 x\right) + x^{2} + 2 x$$

 2             / 2      \
x  + 2*x - a*b*\x  + 2*x/

$$- a b \left(x^{2} + 2 x\right) + x^{2} + 2 x$$

x^2 + 2*x - a*b*(x^2 + 2*x)

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

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Integral of (x^2+y)+(2x-y) dy

Limits of integration:

The graph:

Piecewise:

The solution