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

Lösung

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

6 \frac{( 1 + x ^{\frac{1}{6}} ) ^{3}}{3} - \frac{3 ( 1 + x ^{\frac{1}{6}} ) ^{2}}{2} + 3 (1 + x^{\frac{1}{6}}) - ln 1 + x^{\frac{1}{6}} + C

Schritte zur Lösung

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

Wende U-Substitution an

= \int \frac{6 u ^{3}}{1 + u} d u

Entferne die Konstante:

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

= 6 \cdot \int \frac{u ^{3}}{1 + u} d u

Wende U-Substitution an

= 6 \cdot \int \frac{( v - 1 ) ^{3}}{v} d v

Multipliziere aus

\frac{( v - 1 ) ^{3}}{v} : v^{2} - 3 v + 3 - \frac{1}{v}

= 6 \cdot \int v^{2} - 3 v + 3 - \frac{1}{v} d v

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 6 (\int v^{2} d v - \int 3 v d v + \int 3 d v - \int \frac{1}{v} d v)

\int v^{2} d v = \frac{v ^{3}}{3}

\int 3 v d v = \frac{3 v ^{2}}{2}

\int 3 d v = 3 v

\int \frac{1}{v} d v = ln ∣ v ∣

= 6 (\frac{v ^{3}}{3} - \frac{3 v ^{2}}{2} + 3 v - ln ∣ v ∣)

Ersetze zurück

= 6 \frac{( 1 + x ^{\frac{1}{6}} ) ^{3}}{3} - \frac{3 ( 1 + x ^{\frac{1}{6}} ) ^{2}}{2} + 3 (1 + x^{\frac{1}{6}}) - ln 1 + x^{\frac{1}{6}}

Füge eine Konstante zur Lösung hinzu

= 6 \frac{( 1 + x ^{\frac{1}{6}} ) ^{3}}{3} - \frac{3 ( 1 + x ^{\frac{1}{6}} ) ^{2}}{2} + 3 (1 + x^{\frac{1}{6}}) - ln 1 + x^{\frac{1}{6}} + C

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

\int_{0}^{x} 1 + t^{2} d t

\frac{d}{d x} (4 x^{\frac{1}{4}})

x \to 3 lim (x - 1)

x \to 4 + lim (\frac{3}{( 4 - x ) ^{3}})