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Beliebt Rechnen >

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

Lösung

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

Lösung

\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}{tan ( x ) - 1} d x

Wende U-Substitution an

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

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

Ermittle den Partialbruch von

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

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

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

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

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

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

Setze in

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

ein

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

Vereinfache

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

Graph

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