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Identical expressions
(x²-y²)/(x²+y²)²
(x² minus y²) divide by (x² plus y²)²
x²-y²/x²+y²²
(x²-y²) divide by (x²+y²)²
(x²-y²)/(x²+y²)²dx
Similar expressions
(x²+y²)/(x²+y²)²
(x²-y²)/(x²-y²)²

Integral of (x²-y²)/(x²+y²)² dx

The solution

You have entered [src]

  1              
  /              
 |               
 |    2    2     
 |   x  - y      
 |  ---------- dx
 |           2   
 |  / 2    2\    
 |  \x  + y /    
 |               
/                
0

$$\int\limits_{0}^{1} \frac{x^{2} - y^{2}}{\left(x^{2} + y^{2}\right)^{2}}\, dx$$

Simplify

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\frac{x^{2} - y^{2}}{\left(x^{2} + y^{2}\right)^{2}} = - \frac{2 y^{2}}{\left(x^{2} + y^{2}\right)^{2}} + \frac{1}{x^{2} + y^{2}}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{2 y^{2}}{\left(x^{2} + y^{2}\right)^{2}}\right)\, dx = - 2 y^{2} \int \frac{1}{\left(x^{2} + y^{2}\right)^{2}}\, dx$
    1. Don't know the steps in finding this integral.
      
      But the integral is
      
      $\frac{x}{2 x^{2} y^{2} + 2 y^{4}} + \frac{- \frac{i \log{\left(x - i y \right)}}{4} + \frac{i \log{\left(x + i y \right)}}{4}}{y^{3}}$
    So, the result is: $- 2 y^{2} \left(\frac{x}{2 x^{2} y^{2} + 2 y^{4}} + \frac{- \frac{i \log{\left(x - i y \right)}}{4} + \frac{i \log{\left(x + i y \right)}}{4}}{y^{3}}\right)$
  1. The integral of $\frac{1}{x^{2} + 1}$ is $\frac{\operatorname{atan}{\left(\frac{x}{\sqrt{y^{2}}} \right)}}{\sqrt{y^{2}}}$ .
  The result is: $- 2 y^{2} \left(\frac{x}{2 x^{2} y^{2} + 2 y^{4}} + \frac{- \frac{i \log{\left(x - i y \right)}}{4} + \frac{i \log{\left(x + i y \right)}}{4}}{y^{3}}\right) + \frac{\operatorname{atan}{\left(\frac{x}{\sqrt{y^{2}}} \right)}}{\sqrt{y^{2}}}$
Method #2
1. Rewrite the integrand:
  
  $\frac{x^{2} - y^{2}}{\left(x^{2} + y^{2}\right)^{2}} = \frac{x^{2} - y^{2}}{x^{4} + 2 x^{2} y^{2} + y^{4}}$
2. Rewrite the integrand:
  
  $\frac{x^{2} - y^{2}}{x^{4} + 2 x^{2} y^{2} + y^{4}} = - \frac{2 y^{2}}{\left(x^{2} + y^{2}\right)^{2}} + \frac{1}{x^{2} + y^{2}}$
3. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{2 y^{2}}{\left(x^{2} + y^{2}\right)^{2}}\right)\, dx = - 2 y^{2} \int \frac{1}{\left(x^{2} + y^{2}\right)^{2}}\, dx$
    1. Don't know the steps in finding this integral.
      
      But the integral is
      
      $\frac{x}{2 x^{2} y^{2} + 2 y^{4}} + \frac{- \frac{i \log{\left(x - i y \right)}}{4} + \frac{i \log{\left(x + i y \right)}}{4}}{y^{3}}$
    So, the result is: $- 2 y^{2} \left(\frac{x}{2 x^{2} y^{2} + 2 y^{4}} + \frac{- \frac{i \log{\left(x - i y \right)}}{4} + \frac{i \log{\left(x + i y \right)}}{4}}{y^{3}}\right)$
  1. The integral of $\frac{1}{x^{2} + 1}$ is $\frac{\operatorname{atan}{\left(\frac{x}{\sqrt{y^{2}}} \right)}}{\sqrt{y^{2}}}$ .
  The result is: $- 2 y^{2} \left(\frac{x}{2 x^{2} y^{2} + 2 y^{4}} + \frac{- \frac{i \log{\left(x - i y \right)}}{4} + \frac{i \log{\left(x + i y \right)}}{4}}{y^{3}}\right) + \frac{\operatorname{atan}{\left(\frac{x}{\sqrt{y^{2}}} \right)}}{\sqrt{y^{2}}}$
Method #3
1. Rewrite the integrand:
  
  $\frac{x^{2} - y^{2}}{\left(x^{2} + y^{2}\right)^{2}} = \frac{x^{2}}{x^{4} + 2 x^{2} y^{2} + y^{4}} - \frac{y^{2}}{x^{4} + 2 x^{2} y^{2} + y^{4}}$
2. Integrate term-by-term:
  1. Rewrite the integrand:
    
    $\frac{x^{2}}{x^{4} + 2 x^{2} y^{2} + y^{4}} = - \frac{y^{2}}{\left(x^{2} + y^{2}\right)^{2}} + \frac{1}{x^{2} + y^{2}}$
  2. Integrate term-by-term:
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{y^{2}}{\left(x^{2} + y^{2}\right)^{2}}\right)\, dx = - y^{2} \int \frac{1}{\left(x^{2} + y^{2}\right)^{2}}\, dx$
      
      Don't know the steps in finding this integral.
      
      But the integral is
      
      $\frac{x}{2 x^{2} y^{2} + 2 y^{4}} + \frac{- \frac{i \log{\left(x - i y \right)}}{4} + \frac{i \log{\left(x + i y \right)}}{4}}{y^{3}}$
      
      So, the result is: $- y^{2} \left(\frac{x}{2 x^{2} y^{2} + 2 y^{4}} + \frac{- \frac{i \log{\left(x - i y \right)}}{4} + \frac{i \log{\left(x + i y \right)}}{4}}{y^{3}}\right)$
    1. The integral of $\frac{1}{x^{2} + 1}$ is $\frac{\operatorname{atan}{\left(\frac{x}{\sqrt{y^{2}}} \right)}}{\sqrt{y^{2}}}$ .
    The result is: $- y^{2} \left(\frac{x}{2 x^{2} y^{2} + 2 y^{4}} + \frac{- \frac{i \log{\left(x - i y \right)}}{4} + \frac{i \log{\left(x + i y \right)}}{4}}{y^{3}}\right) + \frac{\operatorname{atan}{\left(\frac{x}{\sqrt{y^{2}}} \right)}}{\sqrt{y^{2}}}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{y^{2}}{x^{4} + 2 x^{2} y^{2} + y^{4}}\right)\, dx = - y^{2} \int \frac{1}{x^{4} + 2 x^{2} y^{2} + y^{4}}\, dx$
    1. Don't know the steps in finding this integral.
      
      But the integral is
      
      $\frac{x}{2 x^{2} y^{2} + 2 y^{4}} + \frac{- \frac{i \log{\left(x - i y \right)}}{4} + \frac{i \log{\left(x + i y \right)}}{4}}{y^{3}}$
    So, the result is: $- y^{2} \left(\frac{x}{2 x^{2} y^{2} + 2 y^{4}} + \frac{- \frac{i \log{\left(x - i y \right)}}{4} + \frac{i \log{\left(x + i y \right)}}{4}}{y^{3}}\right)$
  The result is: $- 2 y^{2} \left(\frac{x}{2 x^{2} y^{2} + 2 y^{4}} + \frac{- \frac{i \log{\left(x - i y \right)}}{4} + \frac{i \log{\left(x + i y \right)}}{4}}{y^{3}}\right) + \frac{\operatorname{atan}{\left(\frac{x}{\sqrt{y^{2}}} \right)}}{\sqrt{y^{2}}}$
Now simplify:

$\frac{y \left(x^{2} + y^{2}\right) \operatorname{atan}{\left(\frac{x}{\sqrt{y^{2}}} \right)} - \frac{\left(2 x y + i \left(x^{2} + y^{2}\right) \left(- \log{\left(x - i y \right)} + \log{\left(x + i y \right)}\right)\right) \sqrt{y^{2}}}{2}}{y \left(x^{2} + y^{2}\right) \sqrt{y^{2}}}$
Add the constant of integration:

$\frac{y \left(x^{2} + y^{2}\right) \operatorname{atan}{\left(\frac{x}{\sqrt{y^{2}}} \right)} - \frac{\left(2 x y + i \left(x^{2} + y^{2}\right) \left(- \log{\left(x - i y \right)} + \log{\left(x + i y \right)}\right)\right) \sqrt{y^{2}}}{2}}{y \left(x^{2} + y^{2}\right) \sqrt{y^{2}}}+ \mathrm{constant}$

The answer is:

$\frac{y \left(x^{2} + y^{2}\right) \operatorname{atan}{\left(\frac{x}{\sqrt{y^{2}}} \right)} - \frac{\left(2 x y + i \left(x^{2} + y^{2}\right) \left(- \log{\left(x - i y \right)} + \log{\left(x + i y \right)}\right)\right) \sqrt{y^{2}}}{2}}{y \left(x^{2} + y^{2}\right) \sqrt{y^{2}}}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

$$-{{x}\over{y^2+x^2}}$$

Simplify

The answer [src]

 -1   
------
     2
1 + y

$$-{{1}\over{y^2+1}}$$

 -1   
------
     2
1 + y

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

Simplify

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of (x²-y²)/(x²+y²)² dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3