integral of 1/(4x^2-9)

Lösung

\int \frac{1}{4 x ^{2} - 9} d x

- \frac{1}{12} (ln \frac{2}{3} x + 1 - ln \frac{2}{3} x - 1) + C

Schritte zur Lösung

\int \frac{1}{4 x ^{2} - 9} d x

Wende integrale Substitution an

= \int \frac{1}{6 ( u ^{2} - 1 )} d u

Entferne die Konstante:

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

= \frac{1}{6} \cdot \int \frac{1}{u ^{2} - 1} d u

Faktorisiere

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

= \frac{1}{6} \cdot \int \frac{1}{- ( - u ^{2} + 1 )} d u

Entferne die Konstante:

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

= \frac{1}{6} (- \int \frac{1}{- u ^{2} + 1} d u)

Nutze das gemeinsame Integral :

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

= \frac{1}{6} (- (\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}))

Setze in

u = \frac{2}{3} x

ein

= \frac{1}{6} (- (\frac{ln \frac{2}{3} x + 1}{2} - \frac{ln \frac{2}{3} x - 1}{2}))

Vereinfache

\frac{1}{6} (- (\frac{ln \frac{2}{3} x + 1}{2} - \frac{ln \frac{2}{3} x - 1}{2})) : - \frac{1}{12} (ln \frac{2}{3} x + 1 - ln \frac{2}{3} x - 1)

= - \frac{1}{12} (ln \frac{2}{3} x + 1 - ln \frac{2}{3} x - 1)

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{12} (ln \frac{2}{3} x + 1 - ln \frac{2}{3} x - 1) + C

\frac{d}{d x} (3 sin (x) cos (x))

d er i v a t i v e f (x) = x^{- 1} - \frac{1}{x}

\frac{d}{d x} (\frac{2}{3 x - 1})

\int y^{2} cos (x) + z^{3} d x

f^{^{''}} (x) = 4 x^{2} + 8 x - 17