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Similar expressions
2*y/(x^2-y^2)+1
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Integral of 2*y/(x^2+y^2)+1 dx

The solution

You have entered [src]

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

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

Integral(2*y/(x^2 + y^2) + 1, (x, 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:
  
  $\int \frac{2 y}{x^{2} + y^{2}}\, dx = 2 y \int \frac{1}{x^{2} + y^{2}}\, dx$
  1. The integral of $\frac{1}{x^{2} + 1}$ is $\frac{\operatorname{atan}{\left(\frac{x}{\sqrt{y^{2}}} \right)}}{\sqrt{y^{2}}}$ .
  So, the result is: $\frac{2 y \operatorname{atan}{\left(\frac{x}{\sqrt{y^{2}}} \right)}}{\sqrt{y^{2}}}$
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 1\, dx = x$
The result is: $x + \frac{2 y \operatorname{atan}{\left(\frac{x}{\sqrt{y^{2}}} \right)}}{\sqrt{y^{2}}}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

                                      /   x   \
                              2*y*atan|-------|
  /                                   |   ____|
 |                                    |  /  2 |
 | /  2*y      \                      \\/  y  /
 | |------- + 1| dx = C + x + -----------------
 | | 2    2    |                      ____     
 | \x  + y     /                     /  2      
 |                                 \/  y       
/

$$2\,\arctan \left({{x}\over{y}}\right)+x$$

Simplify

The answer [src]

1 + I*log(-I*y) + I*log(1 + I*y) - I*log(I*y) - I*log(1 - I*y)

$$2\,\arctan \left({{1}\over{y}}\right)+1$$

1 + I*log(-I*y) + I*log(1 + I*y) - I*log(I*y) - I*log(1 - I*y)

$$i \log{\left(- i y \right)} - i \log{\left(i y \right)} - i \log{\left(- i y + 1 \right)} + i \log{\left(i y + 1 \right)} + 1$$

Simplify

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

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Integral of 2*y/(x^2+y^2)+1 dx

Limits of integration:

The graph:

Piecewise:

The solution