integral of 1/(sec(ax)+1)

Lösung

\int \frac{1}{sec ( a x ) + 1} d x

\frac{1}{a} (2 arctan (tan (\frac{a x}{2})) - tan (\frac{a x}{2})) + C

Schritte zur Lösung

\int \frac{1}{sec ( a x ) + 1} d x

Wende U-Substitution an

= \int \frac{1}{a ( sec ( u ) + 1 )} d u

Entferne die Konstante:

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

= \frac{1}{a} \cdot \int \frac{1}{sec ( u ) + 1} d u

Wende U-Substitution an

= \frac{1}{a} \cdot \int \frac{1 - v ^{2}}{1 + v ^{2}} d v

\frac{1 - v ^{2}}{1 + v ^{2}} = \frac{2}{1 + v ^{2}} - 1

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

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

\int 1 d v = v

= \frac{1}{a} (2 arctan (v) - v)

Ersetze zurück

= \frac{1}{a} (2 arctan (tan (\frac{a x}{2})) - tan (\frac{a x}{2}))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{a} (2 arctan (tan (\frac{a x}{2})) - tan (\frac{a x}{2})) + C