What is the integral of log_{10}(x^2+y^2) ?

The integral of log_{10}(x^2+y^2) is xlog_{10}(y^2+x^2)+2/(ln(10))yarctan(x/y)-(2x)/(ln(10))+C

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integral of log_{10}(x^2+y^2)

integral

lo g_{10} (x^{2} + y^{2})

x lo g_{10} (y^{2} + x^{2}) + \frac{2}{ln ( 10 )} y arctan (\frac{x}{y}) - \frac{2 x}{ln ( 10 )} + C

Solution steps

\int lo g_{10} (x^{2} + y^{2}) d x

Apply Integration By Parts

: x lo g_{10} (y^{2} + x^{2}) - \int \frac{2 x ^{2}}{ln ( 10 ) ( y ^{2} + x ^{2} )} d x

= x lo g_{10} (y^{2} + x^{2}) - \int \frac{2 x ^{2}}{ln ( 10 ) ( y ^{2} + x ^{2} )} d x

\int \frac{2 x ^{2}}{ln ( 10 ) ( y ^{2} + x ^{2} )} d x = - \frac{2}{ln ( 10 )} y arctan (\frac{x}{y}) + \frac{2 x}{ln ( 10 )}

= x lo g_{10} (y^{2} + x^{2}) - (- \frac{2}{ln ( 10 )} y arctan (\frac{x}{y}) + \frac{2 x}{ln ( 10 )})

Simplify

= x lo g_{10} (y^{2} + x^{2}) + \frac{2}{ln ( 10 )} y arctan (\frac{x}{y}) - \frac{2 x}{ln ( 10 )}

Add a constant to the solution

= x lo g_{10} (y^{2} + x^{2}) + \frac{2}{ln ( 10 )} y arctan (\frac{x}{y}) - \frac{2 x}{ln ( 10 )} + C

\frac{( 1 - x ^{2} )}{( 1 + x ^{2} )}

\frac{3}{( x ^{2} + 1 )}

1 3^{x} + 1

arctan (x + 1)

\frac{1}{s ^{3}}

What is the integral of log_{10}(x^2+y^2) ?
The integral of log_{10}(x^2+y^2) is xlog_{10}(y^2+x^2)+2/(ln(10))yarctan(x/y)-(2x)/(ln(10))+C