Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of x^(-1/2)
Integral of e^(2*x+1)
Integral of 1/sqrt(1+u^2)
Integral of -e^x
Identical expressions
(x^ two - two x+y^ two)/((x^ two +y^ two)^2)
(x squared minus 2x plus y squared ) divide by ((x squared plus y squared ) squared )
(x to the power of two minus two x plus y to the power of two) divide by ((x to the power of two plus y to the power of two) squared )
(x2-2x+y2)/((x2+y2)2)
x2-2x+y2/x2+y22
(x²-2x+y²)/((x²+y²)²)
(x to the power of 2-2x+y to the power of 2)/((x to the power of 2+y to the power of 2) to the power of 2)
x^2-2x+y^2/x^2+y^2^2
(x^2-2x+y^2) divide by ((x^2+y^2)^2)
(x^2-2x+y^2)/((x^2+y^2)^2)dx
Similar expressions
(x^2-2x-y^2)/((x^2+y^2)^2)
(x^2-2x+y^2)/((x^2-y^2)^2)
(x^2+2x+y^2)/((x^2+y^2)^2)

Solving integrals/ (x^2-2x+y^2)/((x^2+y^2)^2)

Integral of (x^2-2x+y^2)/((x^2+y^2)^2) dy

The solution

You have entered [src]

  1                 
  /                 
 |                  
 |   2          2   
 |  x  - 2*x + y    
 |  ------------- dy
 |             2    
 |    / 2    2\     
 |    \x  + y /     
 |                  
/                   
0

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

Integral((x^2 - 2*x + y^2)/(x^2 + y^2)^2, (y, 0, 1))

Detail solution

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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

$$\int \frac{y^{2} + \left(x^{2} - 2 x\right)}{\left(x^{2} + y^{2}\right)^{2}}\, dy = C - 2 x \left(\frac{y}{2 x^{4} + 2 x^{2} y^{2}} + \frac{- \frac{i \log{\left(- i x + y \right)}}{4} + \frac{i \log{\left(i x + y \right)}}{4}}{x^{3}}\right) + \frac{\operatorname{atan}{\left(\frac{y}{\sqrt{x^{2}}} \right)}}{\sqrt{x^{2}}}$$

Simplify

The answer [src]

    1      I*(-1 + x)*log(-I*x)   I*(-1 + x)*log(1 + I*x)   I*(-1 + x)*log(I*x)   I*(-1 + x)*log(1 - I*x)
- ------ + -------------------- + ----------------------- - ------------------- - -----------------------
       3              2                        2                       2                       2         
  x + x            2*x                      2*x                     2*x                     2*x

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

    1      I*(-1 + x)*log(-I*x)   I*(-1 + x)*log(1 + I*x)   I*(-1 + x)*log(I*x)   I*(-1 + x)*log(1 - I*x)
- ------ + -------------------- + ----------------------- - ------------------- - -----------------------
       3              2                        2                       2                       2         
  x + x            2*x                      2*x                     2*x                     2*x

-1/(x + x^3) + i*(-1 + x)*log(-i*x)/(2*x^2) + i*(-1 + x)*log(1 + i*x)/(2*x^2) - i*(-1 + x)*log(i*x)/(2*x^2) - i*(-1 + x)*log(1 - i*x)/(2*x^2)

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of (x^2-2x+y^2)/((x^2+y^2)^2) dy

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3