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Similar expressions
(((x*y+1)/x)+y/x)
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Integral of (-((x*y+1)/x)+y/x) dy

The solution

You have entered [src]

  1                   
  /                   
 |                    
 |  /  x*y + 1   y\   
 |  |- ------- + -| dy
 |  \     x      x/   
 |                    
/                     
0

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

Integral(-(x*y + 1)/x + y/x, (y, 0, 1))

Detail solution

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

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

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

The answer is:

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

The answer (Indefinite) [src]

                                         2
  /                                   x*y 
 |                            2   y + ----
 | /  x*y + 1   y\           y         2  
 | |- ------- + -| dy = C + --- - --------
 | \     x      x/          2*x      x    
 |                                        
/

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

Simplify

The answer [src]

  1   1 - x
- - + -----
  x    2*x

$$\frac{1 - x}{2 x} - \frac{1}{x}$$

  1   1 - x
- - + -----
  x    2*x

$$\frac{1 - x}{2 x} - \frac{1}{x}$$

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

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

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Integral of (-((x*y+1)/x)+y/x) dy

Limits of integration:

The graph:

Piecewise:

The solution