integral of (cos(x))/(4-sin^2(x))

Lösung

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

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Wende integrale Substitution an

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

Entferne die Konstante:

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

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

Nutze das gemeinsame Integral :

\int \frac{1}{- v ^{2} + 1} d v = \frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}

= \frac{1}{2} (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2})

Ersetze zurück

= \frac{1}{2} \frac{ln \frac{s i n ( x )}{2} + 1}{2} - \frac{ln \frac{s i n ( x )}{2} - 1}{2}

Vereinfache

\frac{1}{2} \frac{ln \frac{s i n ( x )}{2} + 1}{2} - \frac{ln \frac{s i n ( x )}{2} - 1}{2} : \frac{1}{4} (ln \frac{sin ( x )}{2} + 1 - ln \frac{sin ( x )}{2} - 1)

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

Füge eine Konstante zur Lösung hinzu

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

\frac{d ^{2}}{d x ^{2}} (x y)

\int_{0}^{l n (4)} e^{x} - e^{- 3 x} d x

x \to + 0 lim (\frac{tan ( x ) - x}{x})

\int_{- 2}^{2} (x + 3) 4 - x^{2} d x

d er i v a t i v e f (x) = 2 e^{x^{3}} + 1