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Beliebt Rechnen >

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

Lösung

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

Lösung

- arctan (- 1 + x^{2}) + - 1 + x^{2} + C

Schritte zur Lösung

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

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

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

Vereinfache

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

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

Wende U-Substitution an

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

\frac{u ^{2}}{u ^{2} + 1} = - \frac{1}{u ^{2} + 1} + 1

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

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}{u ^{2} + 1} d u + \int 1 d u

\int \frac{1}{u ^{2} + 1} d u = arctan (u)

\int 1 d u = u

= - arctan (u) + u

Setze in

u = - 1 + x^{2}

ein

= - arctan (- 1 + x^{2}) + - 1 + x^{2}

Füge eine Konstante zur Lösung hinzu

= - arctan (- 1 + x^{2}) + - 1 + x^{2} + C

Graph

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