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

Limits of integration:

from to
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The graph:

from to

Piecewise:

The solution

You have entered [src]
  1                
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 |     / 2    2\   
 |  log\x  + y / dy
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$$\int\limits_{0}^{1} \log{\left(x^{2} + y^{2} \right)}\, dy$$
Integral(log(x^2 + y^2), (y, 0, 1))
Detail solution
  1. Use integration by parts:

    Let and let .

    Then .

    To find :

    1. The integral of a constant is the constant times the variable of integration:

    Now evaluate the sub-integral.

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

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      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:

      1. The integral of a constant is the constant times the variable of integration:

      The result is:

    So, the result is:

  3. Add the constant of integration:


The answer is:

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        
$$\int \log{\left(x^{2} + y^{2} \right)}\, dy = C + \frac{2 x^{2} \operatorname{atan}{\left(\frac{y}{\sqrt{x^{2}}} \right)}}{\sqrt{x^{2}}} + y \log{\left(x^{2} + y^{2} \right)} - 2 y$$
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     /              
$$2 x \left(\frac{i \log{\left(- i x \right)}}{2} - \frac{i \log{\left(i x \right)}}{2}\right) - 2 x \left(\frac{i \log{\left(- i x + 1 \right)}}{2} - \frac{i \log{\left(i x + 1 \right)}}{2}\right) + \log{\left(x^{2} + 1 \right)} - 2$$
-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.