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

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

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The solution

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∫01log⁡(x2+y2) dy\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:

    ∫u⁡dv⁡=u⁡v⁡−∫v⁡du⁡\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

    Let u(y)=log⁡(x2+y2)u{\left(y \right)} = \log{\left(x^{2} + y^{2} \right)} and let dv⁡(y)=1\operatorname{dv}{\left(y \right)} = 1.

    Then du⁡(y)=2yx2+y2\operatorname{du}{\left(y \right)} = \frac{2 y}{x^{2} + y^{2}}.

    To find v(y)v{\left(y \right)}:

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

      ∫1 dy=y\int 1\, dy = y

    Now evaluate the sub-integral.

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

    ∫2y2x2+y2 dy=2∫y2x2+y2 dy\int \frac{2 y^{2}}{x^{2} + y^{2}}\, dy = 2 \int \frac{y^{2}}{x^{2} + y^{2}}\, dy

    1. Rewrite the integrand:

      y2x2+y2=−x2x2+y2+1\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:

        ∫(−x2x2+y2) dy=−x2∫1x2+y2 dy\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 1y2+1\frac{1}{y^{2} + 1} is atan⁡(yx2)x2\frac{\operatorname{atan}{\left(\frac{y}{\sqrt{x^{2}}} \right)}}{\sqrt{x^{2}}}.

        So, the result is: −x2atan⁡(yx2)x2- \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:

        ∫1 dy=y\int 1\, dy = y

      The result is: −x2atan⁡(yx2)x2+y- \frac{x^{2} \operatorname{atan}{\left(\frac{y}{\sqrt{x^{2}}} \right)}}{\sqrt{x^{2}}} + y

    So, the result is: −2x2atan⁡(yx2)x2+2y- \frac{2 x^{2} \operatorname{atan}{\left(\frac{y}{\sqrt{x^{2}}} \right)}}{\sqrt{x^{2}}} + 2 y

  3. Add the constant of integration:

    2x2atan⁡(yx2)x2+ylog⁡(x2+y2)−2y+constant\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:

2x2atan⁡(yx2)x2+ylog⁡(x2+y2)−2y+constant\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|-------|
  /                                                      |   ____|
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 |    / 2    2\                     / 2    2\            \\/  x  /
 | log\x  + y / dy = C - 2*y + y*log\x  + y / + ------------------
 |                                                      ____      
/                                                      /  2       
                                                     \/  x
∫log⁡(x2+y2) dy=C+2x2atan⁡(yx2)x2+ylog⁡(x2+y2)−2y\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     /
2x(ilog⁡(−ix)2−ilog⁡(ix)2)−2x(ilog⁡(−ix+1)2−ilog⁡(ix+1)2)+log⁡(x2+1)−22 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     /
2x(ilog⁡(−ix)2−ilog⁡(ix)2)−2x(ilog⁡(−ix+1)2−ilog⁡(ix+1)2)+log⁡(x2+1)−22 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.

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