integral of (x^3)/(x^2+4)

Lösung

\int \frac{x ^{3}}{x ^{2} + 4} d x

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

Schritte zur Lösung

\int \frac{x ^{3}}{x ^{2} + 4} d x

Wende U-Substitution an

= \int \frac{u - 4}{2 u} 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 - 4}{u} d u

Multipliziere aus

\frac{u - 4}{u} : 1 - \frac{4}{u}

= \frac{1}{2} \cdot \int 1 - \frac{4}{u} 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 1 d u - \int \frac{4}{u} d u)

\int 1 d u = u

\int \frac{4}{u} d u = 4 ln ∣ u ∣

= \frac{1}{2} (u - 4 ln ∣ u ∣)

Setze in

u = x^{2} + 4

ein

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

Füge eine Konstante zur Lösung hinzu

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

l a pl a ce t r an s f or m 5 sin (3 t - \frac{π}{4})

a re a e^{x}, e^{- x}, [0, 1]

\frac{\partial}{\partial x} ((x^{3} - y^{2})^{- 1})

x \to 0 + lim ((1 + x)^{(\frac{1}{x})})

\frac{d}{d x} (25 x^{3} - 6 x^{2} + 7 x - 88)