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(one -5x+7y+6xy)
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(one minus 5x plus 7y plus 6xy)
1-5x+7y+6xy
(1-5x+7y+6xy)dx
Similar expressions
(1+5x+7y+6xy)
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Integral of (1-5x+7y+6xy) dy

The solution

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 2*x                          
  /                           
 |                            
 |  (1 - 5*x + 7*y + 6*x*y) dy
 |                            
/                             
-x

$$\int\limits_{- x}^{2 x} \left(6 x y + \left(7 y + \left(1 - 5 x\right)\right)\right)\, dy$$

Integral(1 - 5*x + 7*y + (6*x)*y, (y, -x, 2*x))

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 6 x y\, dy = 6 x \int y\, dy$
  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: $3 x y^{2}$
1. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 7 y\, dy = 7 \int y\, dy$
    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: $\frac{7 y^{2}}{2}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int \left(1 - 5 x\right)\, dy = y \left(1 - 5 x\right)$
  The result is: $\frac{7 y^{2}}{2} + y \left(1 - 5 x\right)$
The result is: $3 x y^{2} + \frac{7 y^{2}}{2} + y \left(1 - 5 x\right)$
Now simplify:

$\frac{y \left(6 x y - 10 x + 7 y + 2\right)}{2}$
Add the constant of integration:

$\frac{y \left(6 x y - 10 x + 7 y + 2\right)}{2}+ \mathrm{constant}$

The answer is:

$\frac{y \left(6 x y - 10 x + 7 y + 2\right)}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                    2                       
 |                                  7*y                       2
 | (1 - 5*x + 7*y + 6*x*y) dy = C + ---- + y*(1 - 5*x) + 3*x*y 
 |                                   2                         
/

$$\int \left(6 x y + \left(7 y + \left(1 - 5 x\right)\right)\right)\, dy = C + 3 x y^{2} + \frac{7 y^{2}}{2} + y \left(1 - 5 x\right)$$

Simplify

The answer [src]

                   2            
3*x*(1 - 5*x) + 3*x *(7/2 + 3*x)

$$3 x^{2} \left(3 x + \frac{7}{2}\right) + 3 x \left(1 - 5 x\right)$$

                   2            
3*x*(1 - 5*x) + 3*x *(7/2 + 3*x)

$$3 x^{2} \left(3 x + \frac{7}{2}\right) + 3 x \left(1 - 5 x\right)$$

3*x*(1 - 5*x) + 3*x^2*(7/2 + 3*x)

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

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Integral of (1-5x+7y+6xy) dy

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The solution