What is the integral of ln(x^2+y^2) ?

The integral of ln(x^2+y^2) is xln(y^2+x^2)+2yarctan(x/y)-2x+C

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integral of ln(x^2+y^2)

\int ln (x^{2} + y^{2}) d x

x ln (y^{2} + x^{2}) + 2 y arctan (\frac{x}{y}) - 2 x + C

Solution steps

\int ln (x^{2} + y^{2}) d x

Apply Integration By Parts

: x ln (y^{2} + x^{2}) - \int \frac{2 x ^{2}}{y ^{2} + x ^{2}} d x

= x ln (y^{2} + x^{2}) - \int \frac{2 x ^{2}}{y ^{2} + x ^{2}} d x

\int \frac{2 x ^{2}}{y ^{2} + x ^{2}} d x = - 2 y arctan (\frac{x}{y}) + 2 x

= x ln (y^{2} + x^{2}) - (- 2 y arctan (\frac{x}{y}) + 2 x)

Simplify

= x ln (y^{2} + x^{2}) + 2 y arctan (\frac{x}{y}) - 2 x

Add a constant to the solution

= x ln (y^{2} + x^{2}) + 2 y arctan (\frac{x}{y}) - 2 x + C

\int 2 sin (3 t) d t

\int \frac{2}{1 - ( π x ) ^{2}} d x

\frac{d}{d x} (f (x) f (x) (x^{2}))

x \to 2 lim (\frac{x - 4}{x - 2})

x \to - \infty lim (x^{3} + x)

What is the integral of ln(x^2+y^2) ?
The integral of ln(x^2+y^2) is xln(y^2+x^2)+2yarctan(x/y)-2x+C