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Beliebt Rechnen >

integral of (9sqrt(x^2-1))/x

Lösung

\int \frac{9 x ^{2} - 1}{x} d x

Lösung

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

Schritte zur Lösung

\int \frac{9 x ^{2} - 1}{x} d x

Entferne die Konstante:

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

= 9 \cdot \int \frac{x ^{2} - 1}{x} d x

Wende U-Substitution an

= 9 \cdot \int \frac{u ^{2}}{u ^{2} + 1} d u

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

= 9 \cdot \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

= 9 (- \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

= 9 (- arctan (u) + u)

Setze in

u = x^{2} - 1

ein

= 9 (- arctan (x^{2} - 1) + x^{2} - 1)

Füge eine Konstante zur Lösung hinzu

= 9 (- arctan (x^{2} - 1) + x^{2} - 1) + C

Graph

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