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Beliebt Rechnen >

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

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Multipliziere aus

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

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

\int 1 d u = u

\int \frac{1}{u} d u = ln ∣ u ∣

= \frac{1}{2} (u - ln ∣ u ∣)

Setze in

u = 1 + x^{2}

ein

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

Füge eine Konstante zur Lösung hinzu

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

Graph

Beliebte Beispiele

derivative x^{-1/4}+x^{1/4}

d er i v a t i v e x^{- \frac{1}{4}} + x^{\frac{1}{4}}

tangent f(x)=x^2-7,\at x=3

t an g e n t f (x) = x^{2} - 7, at x = 3

fläche 2/x , 2/(x^2),4

a re a \frac{2}{x}, \frac{2}{x ^{2}}, 4

(y-x)e^x+(1+e^x)y^'=0,y(1)=1

(y - x) e^{x} + (1 + e^{x}) y^{^{'}} = 0, y (1) = 1

implicit (dy)/(dx),50x^{1/3}y^{2/3}=600

im pl i c i t \frac{d y}{d x}, 50 x^{\frac{1}{3}} y^{\frac{2}{3}} = 600