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Beliebt Rechnen >

integral of arcsin(1/x)

Lösung

\int arcsin (\frac{1}{x}) d x

Lösung

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

Schritte zur Lösung

\int arcsin (\frac{1}{x}) d x

Wende U-Substitution an

= \int - \frac{arcsin ( u )}{u ^{2}} d u

Entferne die Konstante:

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

= - \int \frac{arcsin ( u )}{u ^{2}} d u

Wende die partielle Integration an

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

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

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

Setze in

u = \frac{1}{x}

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

Graph

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