integral of (x+sin(x))/(cos(x)+1)

Lösung

\int \frac{x + sin ( x )}{cos ( x ) + 1} d x

x tan (\frac{x}{2}) + 2 ln cos (\frac{x}{2}) - ln ∣ cos (x) + 1 ∣ + C

Schritte zur Lösung

\int \frac{x + sin ( x )}{cos ( x ) + 1} d x

Multipliziere aus

\frac{x + sin ( x )}{cos ( x ) + 1} : \frac{x}{cos ( x ) + 1} + \frac{sin ( x )}{cos ( x ) + 1}

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{x}{cos ( x ) + 1} d x + \int \frac{sin ( x )}{cos ( x ) + 1} d x

\int \frac{x}{cos ( x ) + 1} d x = x tan (\frac{x}{2}) + 2 ln cos (\frac{x}{2})

\int \frac{sin ( x )}{cos ( x ) + 1} d x = - ln ∣ cos (x) + 1 ∣

= x tan (\frac{x}{2}) + 2 ln cos (\frac{x}{2}) - ln ∣ cos (x) + 1 ∣

Füge eine Konstante zur Lösung hinzu

= x tan (\frac{x}{2}) + 2 ln cos (\frac{x}{2}) - ln ∣ cos (x) + 1 ∣ + C

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