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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{1}{4} x 2 x^{2} - 9 + \frac{9}{4 2} ln (\frac{2 x ^{2} - 9 + 2 x}{3}) + C

Schritte zur Lösung

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

Trigonometrische Substitution anwenden

= \int \frac{9 sec ^{3} ( u )}{2 2} d u

Entferne die Konstante:

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

= \frac{9}{2 2} \cdot \int sec^{3} (u) d u

Wende integrale Reduktion an

= \frac{9}{2 2} (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} \cdot \int sec (u) d u)

\int sec (u) d u = ln ∣ tan (u) + sec (u) ∣

= \frac{9}{2 2} (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

Setze in

u = arcsec (\frac{2}{3} x)

ein

= \frac{9}{2 2} \frac{sec ^{2} ( arcsec ( \frac{2}{3} x ) ) sin ( arcsec ( \frac{2}{3} x ) )}{2} + \frac{1}{2} ln tan (arcsec (\frac{2}{3} x)) + sec (arcsec (\frac{2}{3} x))

Vereinfache

\frac{9}{2 2} \frac{sec ^{2} ( arcsec ( \frac{2}{3} x ) ) sin ( arcsec ( \frac{2}{3} x ) )}{2} + \frac{1}{2} ln tan (arcsec (\frac{2}{3} x)) + sec (arcsec (\frac{2}{3} x)) : \frac{1}{4} x 2 x^{2} - 9 + \frac{9}{4 2} ln (\frac{2 x ^{2} - 9 + 2 x}{3})

= \frac{1}{4} x 2 x^{2} - 9 + \frac{9}{4 2} ln (\frac{2 x ^{2} - 9 + 2 x}{3})

Füge eine Konstante zur Lösung hinzu

= \frac{1}{4} x 2 x^{2} - 9 + \frac{9}{4 2} ln (\frac{2 x ^{2} - 9 + 2 x}{3}) + C

Graph

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