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ln(x^2-y^2)

Integral of ln(x^2+y^2) da

The solution

You have entered [src]

  1                
  /                
 |                 
 |     / 2    2\   
 |  log\x  + y / dy
 |                 
/                  
0

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

Integral(log(x^2 + y^2), (y, 0, 1))

Detail solution

Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(y \right)} = \log{\left(x^{2} + y^{2} \right)}$ and let $\operatorname{dv}{\left(y \right)} = 1$ .

Then $\operatorname{du}{\left(y \right)} = \frac{2 y}{x^{2} + y^{2}}$ .

To find $v{\left(y \right)}$ :
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 1\, dy = y$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int \frac{2 y^{2}}{x^{2} + y^{2}}\, dy = 2 \int \frac{y^{2}}{x^{2} + y^{2}}\, dy$
1. Rewrite the integrand:
  
  $\frac{y^{2}}{x^{2} + y^{2}} = - \frac{x^{2}}{x^{2} + y^{2}} + 1$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{x^{2}}{x^{2} + y^{2}}\right)\, dy = - x^{2} \int \frac{1}{x^{2} + y^{2}}\, dy$
    1. The integral of $\frac{1}{y^{2} + 1}$ is $\frac{\operatorname{atan}{\left(\frac{y}{\sqrt{x^{2}}} \right)}}{\sqrt{x^{2}}}$ .
    So, the result is: $- \frac{x^{2} \operatorname{atan}{\left(\frac{y}{\sqrt{x^{2}}} \right)}}{\sqrt{x^{2}}}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int 1\, dy = y$
  The result is: $- \frac{x^{2} \operatorname{atan}{\left(\frac{y}{\sqrt{x^{2}}} \right)}}{\sqrt{x^{2}}} + y$
So, the result is: $- \frac{2 x^{2} \operatorname{atan}{\left(\frac{y}{\sqrt{x^{2}}} \right)}}{\sqrt{x^{2}}} + 2 y$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

                                                   2     /   y   \
                                                2*x *atan|-------|
  /                                                      |   ____|
 |                                                       |  /  2 |
 |    / 2    2\                     / 2    2\            \\/  x  /
 | log\x  + y / dy = C - 2*y + y*log\x  + y / + ------------------
 |                                                      ____      
/                                                      /  2       
                                                     \/  x

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

Simplify

The answer [src]

         /I*log(1 - I*x)   I*log(1 + I*x)\       /I*log(-I*x)   I*log(I*x)\      /     2\
-2 - 2*x*|-------------- - --------------| + 2*x*|----------- - ----------| + log\1 + x /
         \      2                2       /       \     2            2     /

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

         /I*log(1 - I*x)   I*log(1 + I*x)\       /I*log(-I*x)   I*log(I*x)\      /     2\
-2 - 2*x*|-------------- - --------------| + 2*x*|----------- - ----------| + log\1 + x /
         \      2                2       /       \     2            2     /

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

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

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Integral of ln(x^2+y^2) da

Limits of integration:

The graph:

Piecewise:

The solution