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

Lösung

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

3 (\frac{1}{4} ln 3 x - 2 + 1 + \frac{3}{8} ln 4 (x - 2)^{\frac{2}{3}} - 4 3 x - 2 + 8 - \frac{1}{4 7} arctan (\frac{1}{7} (2 3 x - 2 - 1))) + C

Schritte zur Lösung

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

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Ermittle den Partialbruch von

\frac{u ^{2}}{u ^{3} + 2 + u} : \frac{1}{4 ( u + 1 )} + \frac{3 u - 2}{4 ( u ^{2} - u + 2 )}

= 3 \cdot \int \frac{1}{4 ( u + 1 )} + \frac{3 u - 2}{4 ( u ^{2} - u + 2 )} 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

= 3 (\int \frac{1}{4 ( u + 1 )} d u + \int \frac{3 u - 2}{4 ( u ^{2} - u + 2 )} d u)

\int \frac{1}{4 ( u + 1 )} d u = \frac{1}{4} ln ∣ u + 1 ∣

\int \frac{3 u - 2}{4 ( u ^{2} - u + 2 )} d u = \frac{1}{8 7} (37 ln 4 u^{2} - 4 u + 8 - 2 arctan (\frac{2 u - 1}{7}))

= 3 (\frac{1}{4} ln ∣ u + 1 ∣ + \frac{1}{8 7} (37 ln 4 u^{2} - 4 u + 8 - 2 arctan (\frac{2 u - 1}{7})))

Setze in

u = 3 x - 2

ein

= 3 (\frac{1}{4} ln 3 x - 2 + 1 + \frac{1}{8 7} (37 ln 4 (3 x - 2)^{2} - 4 3 x - 2 + 8 - 2 arctan (\frac{2 3 x - 2 - 1}{7})))

Vereinfache

3 (\frac{1}{4} ln 3 x - 2 + 1 + \frac{1}{8 7} (37 ln 4 (3 x - 2)^{2} - 4 3 x - 2 + 8 - 2 arctan (\frac{2 3 x - 2 - 1}{7}))) : 3 (\frac{1}{4} ln 3 x - 2 + 1 + \frac{3}{8} ln 4 (x - 2)^{\frac{2}{3}} - 4 3 x - 2 + 8 - \frac{1}{4 7} arctan (\frac{1}{7} (2 3 x - 2 - 1)))

= 3 (\frac{1}{4} ln 3 x - 2 + 1 + \frac{3}{8} ln 4 (x - 2)^{\frac{2}{3}} - 4 3 x - 2 + 8 - \frac{1}{4 7} arctan (\frac{1}{7} (2 3 x - 2 - 1)))

Füge eine Konstante zur Lösung hinzu

= 3 (\frac{1}{4} ln 3 x - 2 + 1 + \frac{3}{8} ln 4 (x - 2)^{\frac{2}{3}} - 4 3 x - 2 + 8 - \frac{1}{4 7} arctan (\frac{1}{7} (2 3 x - 2 - 1))) + C

\int x^{2} (x^{2} - 25)^{\frac{3}{2}} d x

\int \frac{2 x - 3}{4 x ^{2} - 1} d x

\int (e^{4} - 2^{e}) d x

\int ln (\frac{1}{1 - x}) d x

\int \frac{x}{( x - 1 ) ( x - 2 ) ^{2}} d x