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Beliebt Rechnen >

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

Lösung

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

Lösung

- \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

Graph

Beliebte Beispiele

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\int (2 x + 6) ln (x) d x

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\int \frac{x ^{4} - 1}{x + 1} d x

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\int ln (x^{0}) d x

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