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

Lösung

\int \frac{1}{3 tan ( x )} d x

\frac{1}{4} 2 ln tan^{\frac{2}{3}} (x) + 1 - ln \frac{4 tan ^{\frac{4}{3}} ( x ) - 4 tan ^{\frac{2}{3}} ( x ) + 4}{3} + 23 arctan (\frac{2 tan ^{\frac{2}{3}} ( x ) - 1}{3}) + C

Schritte zur Lösung

\int \frac{1}{3 tan ( x )} d x

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Faktorisiere

\frac{u}{u ^{6} + 1}

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

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Ermittle den Partialbruch von

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

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

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

\int \frac{1}{3 ( v + 1 )} d v = \frac{1}{3} ln ∣ v + 1 ∣

\int \frac{- v + 2}{3 ( v ^{2} - v + 1 )} d v = \frac{1}{6} (- ln \frac{4}{3} + \frac{4 v ^{2} - 4 v}{3} + 23 arctan (\frac{2 v - 1}{3}))

= 3 \cdot \frac{1}{2} (\frac{1}{3} ln ∣ v + 1 ∣ + \frac{1}{6} (- ln \frac{4}{3} + \frac{4 v ^{2} - 4 v}{3} + 23 arctan (\frac{2 v - 1}{3})))

Ersetze zurück

= 3 \cdot \frac{1}{2} \frac{1}{3} ln (3 tan (x))^{2} + 1 + \frac{1}{6} - ln \frac{4}{3} + \frac{4 ( ( 3 tan ( x ) ) ^{2} ) ^{2} - 4 ( 3 tan ( x ) ) ^{2}}{3} + 23 arctan \frac{2 ( 3 tan ( x ) ) ^{2} - 1}{3}

Vereinfache

3 \cdot \frac{1}{2} \frac{1}{3} ln (3 tan (x))^{2} + 1 + \frac{1}{6} - ln \frac{4}{3} + \frac{4 ( ( 3 tan ( x ) ) ^{2} ) ^{2} - 4 ( 3 tan ( x ) ) ^{2}}{3} + 23 arctan \frac{2 ( 3 tan ( x ) ) ^{2} - 1}{3} : \frac{1}{4} 2 ln tan^{\frac{2}{3}} (x) + 1 - ln \frac{4 tan ^{\frac{4}{3}} ( x ) - 4 tan ^{\frac{2}{3}} ( x ) + 4}{3} + 23 arctan (\frac{2 tan ^{\frac{2}{3}} ( x ) - 1}{3})

= \frac{1}{4} 2 ln tan^{\frac{2}{3}} (x) + 1 - ln \frac{4 tan ^{\frac{4}{3}} ( x ) - 4 tan ^{\frac{2}{3}} ( x ) + 4}{3} + 23 arctan (\frac{2 tan ^{\frac{2}{3}} ( x ) - 1}{3})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4} 2 ln tan^{\frac{2}{3}} (x) + 1 - ln \frac{4 tan ^{\frac{4}{3}} ( x ) - 4 tan ^{\frac{2}{3}} ( x ) + 4}{3} + 23 arctan (\frac{2 tan ^{\frac{2}{3}} ( x ) - 1}{3}) + C

\int x (\frac{sin ( x )}{cos ( x )})^{2} d x

\int (x^{5} - 3 x^{4} + 6 x^{3} + 3) d x

\int 2 + e^{x} d x

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

\int \frac{1}{u ^{7}} d u