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Beliebt Rechnen >

integral of (e^{-ln(1-x^2)})/(x^2)

Lösung

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

Lösung

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

Schritte zur Lösung

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

Vereinfache

\frac{e ^{- l n (1 - x^{2})}}{x ^{2}} : \frac{1}{x ^{2} ( 1 - x ^{2} )}

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

Ermittle den Partialbruch von

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

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

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

\int \frac{1}{2 ( x + 1 )} d x = \frac{1}{2} ln ∣ x + 1 ∣

\int \frac{1}{2 ( x - 1 )} d x = \frac{1}{2} ln ∣ x - 1 ∣

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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