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

\int y^{- 1} + \frac{3}{x} d x

\int (6 x^{2}) (x^{\frac{1}{4}}) d x

\int \frac{1}{6 x + 2} d x

\int \frac{8}{7} 7 x d x

\int 6 x e^{x} d x