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

Lösung

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

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

Schritte zur Lösung

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

a^{2} - a^{2} x^{2} = a^{2} 1 - x^{2}

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

Entferne die Konstante:

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

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

Wende U-Substitution an

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

Setze in

u = 1 - x^{2}

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

\frac{d y}{d x} = \frac{( y ^{3} )}{x ^{2}}

\int \frac{x ^{2} + 1}{3 x} d x

\frac{\partial}{\partial y} ((A^{2} x^{2} + B^{2} y^{2})^{4})

\int \frac{e ^{\frac{4}{x}}}{x ^{2}} d x

\int \frac{1}{x ^{2}} - 3 x d x