integral of 1/(1-cot(x))

Lösung

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

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

Schritte zur Lösung

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \int \frac{sin ( x )}{sin ( x ) - cos ( x )} d x

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Ermittle den Partialbruch von

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

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

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

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

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

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

Setze in

u = tan (\frac{x}{2})

ein

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

Vereinfache

4 (\frac{1}{8} ln tan^{2} (\frac{x}{2}) + 2 tan (\frac{x}{2}) - 1 + \frac{1}{4} (- \frac{1}{2} ln tan^{2} (\frac{x}{2}) + 1 + arctan (tan (\frac{x}{2})))) : \frac{1}{2} ln tan^{2} (\frac{x}{2}) + 2 tan (\frac{x}{2}) - 1 - \frac{1}{2} ln (sec^{2} (\frac{x}{2})) + arctan (tan (\frac{x}{2}))

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

Füge eine Konstante zur Lösung hinzu

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

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

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

\int (\frac{1}{5 x}) d x

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

\int sinh^{2} (t) d t