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

Lösung

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

Schritte zur Lösung

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

Wende die partielle Integration an

= 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 )})

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

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