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y^2-xy
Identical expressions
y^ two -xy
y squared minus xy
y to the power of two minus xy
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y²-xy
y to the power of 2-xy
y^2-xydx
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y^2+xy

Integral of y^2-xy dy

The solution

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       3/2                   
      x                      
        /                    
       |                     
       |        / 2      \   
       |        \y  - x*y/ dy
       |                     
      /                      
            ___              
-3*x + 10*\/ x

$$\int\limits_{10 \sqrt{x} - 3 x}^{x^{\frac{3}{2}}} \left(- x y + y^{2}\right)\, dy$$

Integral(y^2 - x*y, (y, -3*x + 10*sqrt(x), x^(3/2)))

Detail solution

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The answer [src]

                        3                             2
   4   /            ___\     9/2     /            ___\ 
  x    \-3*x + 10*\/ x /    x      x*\-3*x + 10*\/ x / 
- -- - ------------------ + ---- + --------------------
  2            3             3              2

$$\frac{x^{\frac{9}{2}}}{3} - \frac{x^{4}}{2} + \frac{x \left(10 \sqrt{x} - 3 x\right)^{2}}{2} - \frac{\left(10 \sqrt{x} - 3 x\right)^{3}}{3}$$

                        3                             2
   4   /            ___\     9/2     /            ___\ 
  x    \-3*x + 10*\/ x /    x      x*\-3*x + 10*\/ x / 
- -- - ------------------ + ---- + --------------------
  2            3             3              2

$$\frac{x^{\frac{9}{2}}}{3} - \frac{x^{4}}{2} + \frac{x \left(10 \sqrt{x} - 3 x\right)^{2}}{2} - \frac{\left(10 \sqrt{x} - 3 x\right)^{3}}{3}$$

-x^4/2 - (-3*x + 10*sqrt(x))^3/3 + x^(9/2)/3 + x*(-3*x + 10*sqrt(x))^2/2

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

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Integral of y^2-xy dy

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The solution