integral of (x^2)/(sqrt((x^2+100)^3))

Lösung

\int \frac{x ^{2}}{( x ^{2} + 100 ) ^{3}} d x

\frac{x ^{3}}{100 100 + x ^{2}} - \frac{1}{100} x x^{2} + 100 + ln (\frac{1}{10} x + x^{2} + 100) + C

Schritte zur Lösung

\int \frac{x ^{2}}{( x ^{2} + 100 ) ^{3}} d x

Wende die partielle Integration an

= \frac{x ^{3}}{100 100 + x ^{2}} - \int \frac{x ^{2}}{50 x ^{2} + 100} d x

\int \frac{x ^{2}}{50 x ^{2} + 100} d x = \frac{1}{100} x x^{2} + 100 - ln (\frac{1}{10} x + x^{2} + 100)

= \frac{x ^{3}}{100 100 + x ^{2}} - (\frac{1}{100} x x^{2} + 100 - ln (\frac{1}{10} x + x^{2} + 100))

Vereinfache

= \frac{x ^{3}}{100 100 + x ^{2}} - \frac{1}{100} x x^{2} + 100 + ln (\frac{1}{10} x + x^{2} + 100)

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{3}}{100 100 + x ^{2}} - \frac{1}{100} x x^{2} + 100 + ln (\frac{1}{10} x + x^{2} + 100) + C

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