integral of 1/(1-tan(6x))

Lösung

\int \frac{1}{1 - tan ( 6 x )} d x

\frac{1}{12} (- ln - 1 + tan^{2} (3 x) + 2 tan (3 x) + ln (sec^{2} (3 x)) + 2 arctan (tan (3 x))) + C

Schritte zur Lösung

\int \frac{1}{1 - tan ( 6 x )} d x

Wende U-Substitution an

= \int \frac{1}{6 ( 1 - tan ( u ) )} d u

Entferne die Konstante:

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

= \frac{1}{6} \cdot \int \frac{1}{1 - tan ( u )} d u

Wende U-Substitution an

= \frac{1}{6} \cdot \int \frac{2 ( 1 - v ^{2} )}{( 1 - v ^{2} - 2 v ) ( 1 + v ^{2} )} d v

Entferne die Konstante:

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

= \frac{1}{6} \cdot 2 \cdot \int \frac{1 - v ^{2}}{( 1 - v ^{2} - 2 v ) ( 1 + v ^{2} )} d v

Ermittle den Partialbruch von

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

= \frac{1}{6} \cdot 2 \cdot \int \frac{- v - 1}{2 ( v ^{2} + 2 v - 1 )} + \frac{v + 1}{2 ( v ^{2} + 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

= \frac{1}{6} \cdot 2 (\int \frac{- v - 1}{2 ( v ^{2} + 2 v - 1 )} d v + \int \frac{v + 1}{2 ( v ^{2} + 1 )} d v)

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

\int \frac{v + 1}{2 ( v ^{2} + 1 )} d v = \frac{1}{2} (\frac{1}{2} ln v^{2} + 1 + arctan (v))

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

Ersetze zurück

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

Vereinfache

\frac{1}{6} \cdot 2 (- \frac{1}{4} ln tan^{2} (\frac{6 x}{2}) + 2 tan (\frac{6 x}{2}) - 1 + \frac{1}{2} (\frac{1}{2} ln tan^{2} (\frac{6 x}{2}) + 1 + arctan (tan (\frac{6 x}{2})))) : \frac{1}{12} (- ln - 1 + tan^{2} (3 x) + 2 tan (3 x) + ln (sec^{2} (3 x)) + 2 arctan (tan (3 x)))

= \frac{1}{12} (- ln - 1 + tan^{2} (3 x) + 2 tan (3 x) + ln (sec^{2} (3 x)) + 2 arctan (tan (3 x)))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{12} (- ln - 1 + tan^{2} (3 x) + 2 tan (3 x) + ln (sec^{2} (3 x)) + 2 arctan (tan (3 x))) + C