integral from-1/3 to 1/3 of 1/(1-x^2)

Lösung

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

ln (2)

Dezimale

0.69314 \dots

Schritte zur Lösung

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

Nutze das gemeinsame Integral :

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

= [\frac{ln ∣ x + 1 ∣}{2} - \frac{ln ∣ x - 1 ∣}{2}]_{- \frac{1}{3}}^{\frac{1}{3}}

Vereinfache

[\frac{ln ∣ x + 1 ∣}{2} - \frac{ln ∣ x - 1 ∣}{2}]_{- \frac{1}{3}}^{\frac{1}{3}} : [\frac{1}{2} (ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣)]_{- \frac{1}{3}}^{\frac{1}{3}}

= [\frac{1}{2} (ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣)]_{- \frac{1}{3}}^{\frac{1}{3}}

Berechne die Grenzen:

\frac{- 2 ln ( \frac{2}{3} ) + 2 ln ( \frac{4}{3} )}{2}

= \frac{- 2 ln ( \frac{2}{3} ) + 2 ln ( \frac{4}{3} )}{2}

Vereinfache

= ln (2)

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