integral of (sin(x))/(sqrt(4+cos^2(x)))

Lösung

\int \frac{sin ( x )}{4 + cos ^{2} ( x )} d x

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

Schritte zur Lösung

\int \frac{sin ( x )}{4 + cos ^{2} ( x )} d x

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Trigonometrische Substitution anwenden

= - \int sec (v) d v

Nutze das gemeinsame Integral :

\int sec (v) d v = ln ∣ tan (v) + sec (v) ∣

= - ln ∣ tan (v) + sec (v) ∣

Ersetze zurück

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

= - ln (\frac{1}{2} cos (x) + 4 + cos^{2} (x)) + C