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

Lösung

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

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Multipliziere aus

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

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

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

\int \frac{1}{u} d u = ln ∣ u ∣

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

= \frac{1}{2} (ln ∣ u ∣ - (- \frac{1}{u}))

Setze in

u = x^{2} + 1

ein

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

x \to - 5 lim (\frac{x - 5}{x + 5})

a re a x^{2}, 9

x y^{^{'}} + y = 50 x

d er i v a t i v e f (x) = \frac{8 x}{e ^{x}}

\frac{d x}{d t} = 5 x (7 - x)