integral of (x^2)/(\sqrt[3]{5x-1)}

Lösung

\int \frac{x ^{2}}{3 5 x - 1} d x

\frac{1}{5000} (375 x^{2} (5 x - 1)^{\frac{2}{3}} + 90 x (5 x - 1)^{\frac{2}{3}} + 27 (5 x - 1)^{\frac{2}{3}}) + C

Schritte zur Lösung

\int \frac{x ^{2}}{3 5 x - 1} d x

Wende U-Substitution an

= \int \frac{( u + 1 ) ^{2}}{125 3 u} d u

Entferne die Konstante:

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

= \frac{1}{125} \cdot \int \frac{( u + 1 ) ^{2}}{3 u} d u

Wende die partielle Integration an

= \frac{1}{125} (u^{\frac{2}{3}} (u + 1)^{2} - \int \frac{6 3 u u ^{\frac{5}{3}} + 6 3 u u ^{\frac{2}{3}} - u ^{2} - 2 u - 1}{3 u ^{\frac{1}{3}}} d u)

\int \frac{6 3 u u ^{\frac{5}{3}} + 6 3 u u ^{\frac{2}{3}} - u ^{2} - 2 u - 1}{3 u ^{\frac{1}{3}}} d u = \frac{1}{40} u^{\frac{2}{3}} (25 u^{2} + 32 u - 20)

= \frac{1}{125} (u^{\frac{2}{3}} (u + 1)^{2} - \frac{1}{40} u^{\frac{2}{3}} (25 u^{2} + 32 u - 20))

Setze in

u = 5 x - 1

ein

= \frac{1}{125} ((5 x - 1)^{\frac{2}{3}} (5 x - 1 + 1)^{2} - \frac{1}{40} (5 x - 1)^{\frac{2}{3}} (25 (5 x - 1)^{2} + 32 (5 x - 1) - 20))

Vereinfache

\frac{1}{125} ((5 x - 1)^{\frac{2}{3}} (5 x - 1 + 1)^{2} - \frac{1}{40} (5 x - 1)^{\frac{2}{3}} (25 (5 x - 1)^{2} + 32 (5 x - 1) - 20)) : \frac{1}{5000} (375 x^{2} (5 x - 1)^{\frac{2}{3}} + 90 x (5 x - 1)^{\frac{2}{3}} + 27 (5 x - 1)^{\frac{2}{3}})

= \frac{1}{5000} (375 x^{2} (5 x - 1)^{\frac{2}{3}} + 90 x (5 x - 1)^{\frac{2}{3}} + 27 (5 x - 1)^{\frac{2}{3}})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{5000} (375 x^{2} (5 x - 1)^{\frac{2}{3}} + 90 x (5 x - 1)^{\frac{2}{3}} + 27 (5 x - 1)^{\frac{2}{3}}) + C

s l o p e f (x) = 5 x + y = 6

\frac{d}{d x} (6 cos^{2} (x))

x \to - 1 lim (2 x^{2} + 3 x - 1)

n = 1 \sum \infty \frac{8 n}{n !}

\int_{0}^{l n (2)} x e^{- x} d x