integral of 1/(x*sqrt(1-x^2))

Lösung

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

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

Schritte zur Lösung

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

Wende U-Substitution an

= \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

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

Setze in

u = 1 - x^{2}

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

\frac{d}{d x} (a x e^{2 x})

t an g e n t f (x) = x^{3} + x^{2} + (\frac{6}{x}), at x = 1

t an g e n t y = (x - 4)^{2}, at x = 5

\frac{d y}{d t} = - 2 y + 10

\int x^{3} (x^{4} - 1)^{5} d x