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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

sum from n=0 to infinity of (4/9)^{2n+1}

n = 0 \sum \infty (\frac{4}{9})^{2 n + 1}

2xy^'+y=6x,y(4)=24

2 x y^{^{'}} + y = 6 x, y (4) = 24

integral from 0 to 1 of x^2e^{2x}

\int_{0}^{1} x^{2} e^{2 x} d x

derivative of sqrt(2xy)

\frac{d}{d x} (2 x y)

derivative 10x-3x^2

d er i v a t i v e 10 x - 3 x^{2}