de

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Beliebt Rechnen >

integral of 5/(x(x^2+2)^2)

Lösung

\int \frac{5}{x ( x ^{2} + 2 ) ^{2}} d x

Lösung

\frac{5}{2} (\frac{1}{2} ln (x) - \frac{1}{4} ln x^{2} + 2 + \frac{1}{2 ( x ^{2} + 2 )}) + C

Schritte zur Lösung

\int \frac{5}{x ( x ^{2} + 2 ) ^{2}} d x

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 5 \cdot \int \frac{1}{x ( x ^{2} + 2 ) ^{2}} d x

Wende U-Substitution an

= 5 \cdot \int \frac{1}{2 u ( u + 2 ) ^{2}} d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 5 \cdot \frac{1}{2} \cdot \int \frac{1}{u ( u + 2 ) ^{2}} d u

Ermittle den Partialbruch von

\frac{1}{u ( u + 2 ) ^{2}} : \frac{1}{4 u} - \frac{1}{4 ( u + 2 )} - \frac{1}{2 ( u + 2 ) ^{2}}

= 5 \cdot \frac{1}{2} \cdot \int \frac{1}{4 u} - \frac{1}{4 ( u + 2 )} - \frac{1}{2 ( u + 2 ) ^{2}} d u

Wende die Summenregel an:

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

= 5 \cdot \frac{1}{2} (\int \frac{1}{4 u} d u - \int \frac{1}{4 ( u + 2 )} d u - \int \frac{1}{2 ( u + 2 ) ^{2}} d u)

\int \frac{1}{4 u} d u = \frac{1}{4} ln ∣ u ∣

\int \frac{1}{4 ( u + 2 )} d u = \frac{1}{4} ln ∣ u + 2 ∣

\int \frac{1}{2 ( u + 2 ) ^{2}} d u = - \frac{1}{2 ( u + 2 )}

= 5 \cdot \frac{1}{2} (\frac{1}{4} ln ∣ u ∣ - \frac{1}{4} ln ∣ u + 2 ∣ - (- \frac{1}{2 ( u + 2 )}))

Setze in

u = x^{2}

ein

= 5 \cdot \frac{1}{2} (\frac{1}{4} ln x^{2} - \frac{1}{4} ln x^{2} + 2 - (- \frac{1}{2 ( x ^{2} + 2 )}))

Vereinfache

5 \cdot \frac{1}{2} (\frac{1}{4} ln x^{2} - \frac{1}{4} ln x^{2} + 2 - (- \frac{1}{2 ( x ^{2} + 2 )})) : \frac{5}{2} (\frac{1}{2} ln (x) - \frac{1}{4} ln x^{2} + 2 + \frac{1}{2 ( x ^{2} + 2 )})

= \frac{5}{2} (\frac{1}{2} ln (x) - \frac{1}{4} ln x^{2} + 2 + \frac{1}{2 ( x ^{2} + 2 )})

Füge eine Konstante zur Lösung hinzu

= \frac{5}{2} (\frac{1}{2} ln (x) - \frac{1}{4} ln x^{2} + 2 + \frac{1}{2 ( x ^{2} + 2 )}) + C

Graph

Beliebte Beispiele

inverselaplace s/(s^2+s)

in v erse l a pl a ce \frac{s}{s ^{2} + s}

integral of (x^2)/2*1/x

\int \frac{x ^{2}}{2} \cdot \frac{1}{x} d x

laplacetransform 2e^{-5t}sin(2t)

l a pl a ce t r an s f or m 2 e^{- 5 t} sin (2 t)

y^'=3\sqrt[3]{y^2}

y^{^{'}} = 3 3 y^{2}

derivative f(x)=xln(x^2)

d er i v a t i v e f (x) = x ln (x^{2})