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

Lösung

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

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Füge eine Konstante zur Lösung hinzu

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

\frac{d}{d x} (arctan (sin (ln (x^{3}))))

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

s l o p e y = 2 x^{3} - x^{2} + 3

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

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