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x-y/(x+y)^ three
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x-y/(x+y)3
x-y/x+y3
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x-y divide by (x+y)^3
x-y/(x+y)^3dx
Similar expressions
x-y/(x-y)^3
x+y/(x+y)^3

Integral of x-y/(x+y)^3 dx

The solution

You have entered [src]

  1                  
  /                  
 |                   
 |  /       y    \   
 |  |x - --------| dx
 |  |           3|   
 |  \    (x + y) /   
 |                   
/                    
0

$$\int\limits_{0}^{1} \left(x - \frac{y}{\left(x + y\right)^{3}}\right)\, dx$$

Integral(x - y/(x + y)^3, (x, 0, 1))

Detail solution

Integrate term-by-term:
1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int x\, dx = \frac{x^{2}}{2}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{y}{\left(x + y\right)^{3}}\right)\, dx = - y \int \frac{1}{\left(x + y\right)^{3}}\, dx$
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $- \frac{1}{2 x^{2} + 4 x y + 2 y^{2}}$
  So, the result is: $\frac{y}{2 x^{2} + 4 x y + 2 y^{2}}$
The result is: $\frac{x^{2}}{2} + \frac{y}{2 x^{2} + 4 x y + 2 y^{2}}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                                                
 |                          2                      
 | /       y    \          x             y         
 | |x - --------| dx = C + -- + -------------------
 | |           3|          2       2      2        
 | \    (x + y) /               2*x  + 2*y  + 4*x*y
 |                                                 
/

$$\int \left(x - \frac{y}{\left(x + y\right)^{3}}\right)\, dx = C + \frac{x^{2}}{2} + \frac{y}{2 x^{2} + 4 x y + 2 y^{2}}$$

Simplify

The answer [src]

1    1          y       
- - --- + --------------
2   2*y          2      
          2 + 2*y  + 4*y

$$\frac{y}{2 y^{2} + 4 y + 2} + \frac{1}{2} - \frac{1}{2 y}$$

1    1          y       
- - --- + --------------
2   2*y          2      
          2 + 2*y  + 4*y

$$\frac{y}{2 y^{2} + 4 y + 2} + \frac{1}{2} - \frac{1}{2 y}$$

1/2 - 1/(2*y) + y/(2 + 2*y^2 + 4*y)

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

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Integral of x-y/(x+y)^3 dx

Limits of integration:

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Piecewise:

The solution