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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{x ^{3}}{a ^{2} a ^{2} + x ^{2}} - \frac{x x ^{2} + a ^{2}}{a ^{2}} + ln (\frac{x + x ^{2} + a ^{2}}{∣ a ∣}) + C

Schritte zur Lösung

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

(x^{2} + a^{2})^{3} = (x^{2} + a^{2})^{\frac{3}{2}},

angenommen

(x^{2} + a^{2}) \geq 0

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

Wende die partielle Integration an

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

\int \frac{2 x ^{2}}{a ^{2} a ^{2} + x ^{2}} d x = \frac{x x ^{2} + a ^{2}}{a ^{2}} - ln (\frac{x + x ^{2} + a ^{2}}{∣ a ∣})

= \frac{x ^{3}}{a ^{2} a ^{2} + x ^{2}} - (\frac{x x ^{2} + a ^{2}}{a ^{2}} - ln (\frac{x + x ^{2} + a ^{2}}{∣ a ∣}))

Vereinfache

= \frac{x ^{3}}{a ^{2} a ^{2} + x ^{2}} - \frac{x x ^{2} + a ^{2}}{a ^{2}} + ln (\frac{x + x ^{2} + a ^{2}}{∣ a ∣})

Füge eine Konstante zur Lösung hinzu

= \frac{x ^{3}}{a ^{2} a ^{2} + x ^{2}} - \frac{x x ^{2} + a ^{2}}{a ^{2}} + ln (\frac{x + x ^{2} + a ^{2}}{∣ a ∣}) + C

Graph

Beliebte Beispiele

(sin(x)sin(2x)sin(3x))^'

(sin (x) sin (2 x) sin (3 x))^{^{'}}

derivative y=e^xsin(2x)

d er i v a t i v e y = e^{x} sin (2 x)

limit as x approaches 0+of xln(sin(x))

x \to 0 + lim (x ln (sin (x)))

integral of 1/(sqrt(5x-2))

\int \frac{1}{5 x - 2} d x

(dr)/(dθ)=8(cos(θ/3))^3

\frac{d r}{d θ} = 8 (cos (\frac{θ}{3}))^{3}