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Beliebt Rechnen >

integral of (x^4+81)/(x(x^2+9)^2)

Lösung

\int \frac{x ^{4} + 81}{x ( x ^{2} + 9 ) ^{2}} d x

Lösung

\frac{9}{x ^{2} + 9} + ln (x) + C

Schritte zur Lösung

\int \frac{x ^{4} + 81}{x ( x ^{2} + 9 ) ^{2}} d x

Multipliziere aus

\frac{x ^{4} + 81}{x ( x ^{2} + 9 ) ^{2}} : \frac{x ^{4}}{x ( x ^{2} + 9 ) ^{2}} + \frac{81}{x ( x ^{2} + 9 ) ^{2}}

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{x ^{4}}{x ( x ^{2} + 9 ) ^{2}} d x + \int \frac{81}{x ( x ^{2} + 9 ) ^{2}} d x

\int \frac{x ^{4}}{x ( x ^{2} + 9 ) ^{2}} d x = \frac{1}{2} (ln x^{2} + 9 + \frac{9}{x ^{2} + 9})

\int \frac{81}{x ( x ^{2} + 9 ) ^{2}} d x = ln (x) - \frac{1}{2} ln x^{2} + 9 + \frac{9}{2 ( x ^{2} + 9 )}

= \frac{1}{2} (ln x^{2} + 9 + \frac{9}{x ^{2} + 9}) + ln (x) - \frac{1}{2} ln x^{2} + 9 + \frac{9}{2 ( x ^{2} + 9 )}

Vereinfache

\frac{1}{2} (ln x^{2} + 9 + \frac{9}{x ^{2} + 9}) + ln (x) - \frac{1}{2} ln x^{2} + 9 + \frac{9}{2 ( x ^{2} + 9 )} : \frac{9}{x ^{2} + 9} + ln (x)

= \frac{9}{x ^{2} + 9} + ln (x)

Füge eine Konstante zur Lösung hinzu

= \frac{9}{x ^{2} + 9} + ln (x) + C

Graph

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