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integral of 1/(xsqrt(a^2-x^2))

Lösung

\int \frac{1}{x a ^{2} - x ^{2}} d x

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

Schritte zur Lösung

\int \frac{1}{x a ^{2} - x ^{2}} d x

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Faktorisiere

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

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

Entferne die Konstante:

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

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

Wende integrale Substitution an

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

Entferne die Konstante:

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

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

Faktorisiere

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

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

Ersetze zurück

= - - \frac{1}{a} - \frac{ln \frac{a ^{2} - x ^{2}}{a} + 1}{2} - \frac{ln \frac{a ^{2} - x ^{2}}{a} - 1}{2}

Vereinfache

- - \frac{1}{a} - \frac{ln \frac{a ^{2} - x ^{2}}{a} + 1}{2} - \frac{ln \frac{a ^{2} - x ^{2}}{a} - 1}{2} : - \frac{1}{2 a} (ln \frac{a ^{2} - x ^{2}}{a} + 1 - ln \frac{a ^{2} - x ^{2}}{a} - 1)

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

Füge eine Konstante zur Lösung hinzu

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

\frac{d y}{d x} = (y - 1)^{2} + 0.01, y (0) = 1

\frac{d}{d x} ((e^{x} - 1)^{2})

\int \frac{( x + 3 )}{x ^{2} + 11 x + 18} d x

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

d er i v a t i v e y = ln (4 x)