integral of (cos(x))/(sin^2(x)-64)

Lösung

\int \frac{cos ( x )}{sin ^{2} ( x ) - 64} d x

- \frac{1}{16} (ln \frac{sin ( x )}{8} + 1 - ln \frac{sin ( x )}{8} - 1) + C

Schritte zur Lösung

\int \frac{cos ( x )}{sin ^{2} ( x ) - 64} d x

Wende U-Substitution an

= \int \frac{1}{u ^{2} - 64} d u

Wende integrale Substitution an

= \int \frac{1}{8 ( v ^{2} - 1 )} d v

Entferne die Konstante:

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

= \frac{1}{8} \cdot \int \frac{1}{v ^{2} - 1} d v

Faktorisiere

v^{2} - 1 : - (- v^{2} + 1)

= \frac{1}{8} \cdot \int \frac{1}{- ( - v ^{2} + 1 )} d v

Entferne die Konstante:

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

= \frac{1}{8} (- \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}{8} (- (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}))

Ersetze zurück

= \frac{1}{8} - \frac{ln \frac{s i n ( x )}{8} + 1}{2} - \frac{ln \frac{s i n ( x )}{8} - 1}{2}

Vereinfache

\frac{1}{8} - \frac{ln \frac{s i n ( x )}{8} + 1}{2} - \frac{ln \frac{s i n ( x )}{8} - 1}{2} : - \frac{1}{16} (ln \frac{sin ( x )}{8} + 1 - ln \frac{sin ( x )}{8} - 1)

= - \frac{1}{16} (ln \frac{sin ( x )}{8} + 1 - ln \frac{sin ( x )}{8} - 1)

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{16} (ln \frac{sin ( x )}{8} + 1 - ln \frac{sin ( x )}{8} - 1) + C

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