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Beliebt Rechnen >

integral of 1/(1-sec(t))

Lösung

\int \frac{1}{1 - sec ( t )} d t

Lösung

cot (\frac{t}{2}) + t + C

Schritte zur Lösung

\int \frac{1}{1 - sec ( t )} d t

Faktorisiere

1 - sec (t) : - (- 1 + sec (t))

= \int \frac{1}{- ( - 1 + sec ( t ) )} d t

Entferne die Konstante:

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

= - \int \frac{1}{- 1 + sec ( t )} d t

\frac{1}{- 1 + sec ( t )} = \frac{sec ( t )}{- 1 + sec ( t )} - 1

= - \int \frac{sec ( t )}{- 1 + sec ( t )} - 1 d t

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - (\int \frac{sec ( t )}{- 1 + sec ( t )} d t - \int 1 d t)

\int \frac{sec ( t )}{- 1 + sec ( t )} d t = - cot (\frac{t}{2})

\int 1 d t = t

= - (- cot (\frac{t}{2}) - t)

Vereinfache

= cot (\frac{t}{2}) + t

Füge eine Konstante zur Lösung hinzu

= cot (\frac{t}{2}) + t + C

Graph

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