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

Lösung

integral

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

\frac{1}{2} x^{2} lo g_{10} (x)^{2} - \frac{1}{ln ( 10 )} (\frac{1}{2} x^{2} lo g_{10} (x) - \frac{x ^{2}}{4 ln ( 10 )}) + C

Schritte zur Lösung

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

Wende die partielle Integration an

= \frac{1}{2} x^{2} lo g_{10} (x)^{2} - \int \frac{x lo g _{10} ( x )}{ln ( 10 )} d x

\int \frac{x lo g _{10} ( x )}{ln ( 10 )} d x = \frac{1}{ln ( 10 )} (\frac{1}{2} x^{2} lo g_{10} (x) - \frac{x ^{2}}{4 ln ( 10 )})

= \frac{1}{2} x^{2} lo g_{10} (x)^{2} - \frac{1}{ln ( 10 )} (\frac{1}{2} x^{2} lo g_{10} (x) - \frac{x ^{2}}{4 ln ( 10 )})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} x^{2} lo g_{10} (x)^{2} - \frac{1}{ln ( 10 )} (\frac{1}{2} x^{2} lo g_{10} (x) - \frac{x ^{2}}{4 ln ( 10 )}) + C

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