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

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

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1           
  /           
 |            
 |     x      
 |  ------- dy
 |   2    2   
 |  y  + x    
 |            
/             
0
$$\int\limits_{0}^{1} \frac{x}{x^{2} + y^{2}}\, dy$$ 
Integral(x/(y^2 + x^2), (y, 0, 1))
Detail solution

  1. The integral of a constant times a function is the constant times the integral of the function:

    1. The integral of is .

    So, the result is:

  2. Add the constant of integration:

The answer is:

The answer (Indefinite) [src] 
                          /   y   \
                    x*atan|-------|
  /                       |   ____|
 |                        |  /  2 |
 |    x                   \\/  x  /
 | ------- dy = C + ---------------
 |  2    2                 ____    
 | y  + x                 /  2     
 |                      \/  x      
/
$$\int \frac{x}{x^{2} + y^{2}}\, dy = C + \frac{x \operatorname{atan}{\left(\frac{y}{\sqrt{x^{2}}} \right)}}{\sqrt{x^{2}}}$$
Simplify
The answer [src] 
I*log(-I*x)   I*log(1 + I*x)   I*log(I*x)   I*log(1 - I*x)
----------- + -------------- - ---------- - --------------
     2              2              2              2
$$\frac{i \log{\left(- i x \right)}}{2} - \frac{i \log{\left(i x \right)}}{2} - \frac{i \log{\left(- i x + 1 \right)}}{2} + \frac{i \log{\left(i x + 1 \right)}}{2}$$ 
=
= 
I*log(-I*x)   I*log(1 + I*x)   I*log(I*x)   I*log(1 - I*x)
----------- + -------------- - ---------- - --------------
     2              2              2              2
$$\frac{i \log{\left(- i x \right)}}{2} - \frac{i \log{\left(i x \right)}}{2} - \frac{i \log{\left(- i x + 1 \right)}}{2} + \frac{i \log{\left(i x + 1 \right)}}{2}$$ 
i*log(-i*x)/2 + i*log(1 + i*x)/2 - i*log(i*x)/2 - i*log(1 - i*x)/2

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

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