integral of (x^3)/((1+x^2)^3)

Lösung

\int \frac{x ^{3}}{( 1 + x ^{2} ) ^{3}} d x

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

Schritte zur Lösung

\int \frac{x ^{3}}{( 1 + x ^{2} ) ^{3}} d x

Wende U-Substitution an

= \int \frac{u - 1}{2 u ^{3}} d u

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int \frac{u - 1}{u ^{3}} d u

Multipliziere aus

\frac{u - 1}{u ^{3}} : \frac{1}{u ^{2}} - \frac{1}{u ^{3}}

= \frac{1}{2} \cdot \int \frac{1}{u ^{2}} - \frac{1}{u ^{3}} 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

= \frac{1}{2} (\int \frac{1}{u ^{2}} d u - \int \frac{1}{u ^{3}} d u)

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

\int \frac{1}{u ^{3}} d u = - \frac{1}{2 u ^{2}}

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

Setze in

u = 1 + x^{2}

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

x \to \infty lim (\frac{3 x ^{2} - 2 x - 5}{x - 4})

x \to \infty lim (\frac{1}{3 x})

x \to - 2 lim (\frac{x - 2}{x ^{2} - 4})

d er i v a t i v e \frac{5 - x}{( 4 x ^{2} + 1 ) ^{3}}

\frac{d P}{d t} = 5 Pt, P (1) = 6