Integral of x-3y dy

The solution

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    2               
   x                
    /               
   |                
   |    (x - 3*y) dy
   |                
  /                 
   ___              
-\/ x               
-------             
 1 + x

$$\int\limits_{- \frac{\sqrt{x}}{x + 1}}^{x^{2}} \left(x - 3 y\right)\, dy$$

Integral(x - 3*y, (y, -sqrt(x)/(1 + x), x^2))

Detail solution

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The answer [src]

        4     3/2             
 3   3*x     x         3*x    
x  - ---- + ----- + ----------
      2     1 + x            2
                    2*(1 + x)

$$\frac{x^{\frac{3}{2}}}{x + 1} - \frac{3 x^{4}}{2} + x^{3} + \frac{3 x}{2 \left(x + 1\right)^{2}}$$

        4     3/2             
 3   3*x     x         3*x    
x  - ---- + ----- + ----------
      2     1 + x            2
                    2*(1 + x)

$$\frac{x^{\frac{3}{2}}}{x + 1} - \frac{3 x^{4}}{2} + x^{3} + \frac{3 x}{2 \left(x + 1\right)^{2}}$$

x^3 - 3*x^4/2 + x^(3/2)/(1 + x) + 3*x/(2*(1 + x)^2)

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

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Integral of x-3y dy

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