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Beliebt Rechnen >

integral of (x^3)/(x^2+5)

Lösung

\int \frac{x ^{3}}{x ^{2} + 5} d x

Lösung

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

Schritte zur Lösung

\int \frac{x ^{3}}{x ^{2} + 5} d x

Wende U-Substitution an

= \int \frac{u - 5}{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 - 5}{u} d u

Multipliziere aus

\frac{u - 5}{u} : 1 - \frac{5}{u}

= \frac{1}{2} \cdot \int 1 - \frac{5}{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{5}{u} d u)

\int 1 d u = u

\int \frac{5}{u} d u = 5 ln ∣ u ∣

= \frac{1}{2} (u - 5 ln ∣ u ∣)

Setze in

u = x^{2} + 5

ein

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

derivative of (dy/(dx))=4x

\frac{d}{d x} (\frac{d y}{d x}) = 4 x

tangent f(x)=-3e^{-2x}+3

t an g e n t f (x) = - 3 e^{- 2 x} + 3

integral of 1/((x^2+2)^{5/2)}

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

integral of (u-1)/(u^2)

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

integral from 0 to infinity of xe^{-x^2}

\int_{0}^{\infty} x e^{- x^{2}} d x