integral of 1/(sqrt(x)-5\sqrt[3]{x)}

Lösung

\int \frac{1}{x - 5 3 x} d x

6 (\frac{1}{3} x + \frac{5}{2} 3 x + 25 6 x + 125 ln 6 x - 5) + C

Schritte zur Lösung

\int \frac{1}{x - 5 3 x} d x

Wende U-Substitution an

= \int \frac{6 u ^{3}}{u - 5} 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 - 5} d u

Lange Division

\frac{u ^{3}}{u - 5} : u^{2} + 5 u + 25 + \frac{125}{u - 5}

= 6 \cdot \int u^{2} + 5 u + 25 + \frac{125}{u - 5} 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

= 6 (\int u^{2} d u + \int 5 u d u + \int 25 d u + \int \frac{125}{u - 5} d u)

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

\int 5 u d u = \frac{5 u ^{2}}{2}

\int 25 d u = 25 u

\int \frac{125}{u - 5} d u = 125 ln ∣ u - 5 ∣

= 6 (\frac{u ^{3}}{3} + \frac{5 u ^{2}}{2} + 25 u + 125 ln ∣ u - 5 ∣)

Setze in

u = 6 x

ein

= 6 (\frac{( 6 x ) ^{3}}{3} + \frac{5 ( 6 x ) ^{2}}{2} + 25 6 x + 125 ln 6 x - 5)

Vereinfache

6 (\frac{( 6 x ) ^{3}}{3} + \frac{5 ( 6 x ) ^{2}}{2} + 25 6 x + 125 ln 6 x - 5) : 6 (\frac{1}{3} x + \frac{5}{2} 3 x + 25 6 x + 125 ln 6 x - 5)

= 6 (\frac{1}{3} x + \frac{5}{2} 3 x + 25 6 x + 125 ln 6 x - 5)

Füge eine Konstante zur Lösung hinzu

= 6 (\frac{1}{3} x + \frac{5}{2} 3 x + 25 6 x + 125 ln 6 x - 5) + C

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