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Beliebt Rechnen >

integral of 1/(sin(x)+3cos(x)+1)

Lösung

\int \frac{1}{sin ( x ) + 3 cos ( x ) + 1} d x

Lösung

\frac{1}{3} ln tan (\frac{x}{2}) + 1 - \frac{1}{3} ln tan (\frac{x}{2}) - 2 + C

Schritte zur Lösung

\int \frac{1}{sin ( x ) + 3 cos ( x ) + 1} d x

Wende U-Substitution an

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

Ermittle den Partialbruch von

\frac{1}{- u ^{2} + u + 2} : \frac{1}{3 ( u + 1 )} - \frac{1}{3 ( u - 2 )}

= \int \frac{1}{3 ( u + 1 )} - \frac{1}{3 ( u - 2 )} 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

= \int \frac{1}{3 ( u + 1 )} d u - \int \frac{1}{3 ( u - 2 )} d u

\int \frac{1}{3 ( u + 1 )} d u = \frac{1}{3} ln ∣ u + 1 ∣

\int \frac{1}{3 ( u - 2 )} d u = \frac{1}{3} ln ∣ u - 2 ∣

= \frac{1}{3} ln ∣ u + 1 ∣ - \frac{1}{3} ln ∣ u - 2 ∣

Setze in

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

ein

= \frac{1}{3} ln tan (\frac{x}{2}) + 1 - \frac{1}{3} ln tan (\frac{x}{2}) - 2

Füge eine Konstante zur Lösung hinzu

= \frac{1}{3} ln tan (\frac{x}{2}) + 1 - \frac{1}{3} ln tan (\frac{x}{2}) - 2 + C

Graph

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